Complex Variables SOLO HERMELIN 222222222222222222222222

Complex Variables SOLO HERMELIN.ppt

1. 1
Complex Variables
SOLO HERMELIN
Updated: 11.05.07
28.10.12

2. 2
SOLO
Complex Variables
Table of Contents
Set of Numbers – Examples
Fundamentals Operations with Complex Numbers z = x + i y
Axiomatic Foundations of the Complex Number System  R babaz ,,
History of Complex Numbers
Derivatives
Cauchy-Riemann Equations
Harmonic Functions
Orthogonal Families
Singular Points
Complex Line Integrals
Simply and Multiply Connected Regions
Green’s Theorem in the Plane
Consequences of Green’s Theorem in the Plane
Cauchy’s Theorem
Cauchy-Goursat Theorem
Consequences of Cauchy-Goursat Theorem

3. SOLO
Complex Variables
Table of Contents (continue - 1)
Cauchy’s Integral Formulas and Related Theorems
Cauchy’s Integral Formulas
Cauchy’s Integral Formulas for the n Derivative of a Function
Morera’s Theorem (the converse of Cauchy’s theorem)
Cauchy’s Inequality
Liouville’s Theorem
Foundamental Theorem of Algebra
Gauss’ Mean Value Theorem
Maximum Modulus Theorem
Minimum Modulus Theorem
Poisson’s Integral Formulas for a Circle
Poisson’s Integral Formulas for a Half Plane

4. 4
SOLO
Complex Variables
Table of Contents (continue - 2)
Theorems of Convergence of Sequences and Series
Convergence Tests
Cauchy Root Test
D’Alembert or Cauchy Ratio Test
Maclaurin or Euler Integral Test
Kummer’s Test
Raabe’s Test
Gauss’ s Test
Infinite Series, Taylor’s and Laurent Series
Infinite Series of Functions
Absolute Convergence of Series of Functions
Uniformly Convergence of Sequences and Series
Weierstrass M (Majorant) Test
Abel’s Test
Uniformly Convergent Series of Analytic Functions
Taylor’s Series
Laurent’s Series (1843)

5. 5
SOLO
Complex Variables
Table of Contents (continue - 3)
5
The Argument Theorem
Rouché’s Theorem
Foundamental Theorem of Algebra (using Rouché’s Theorem)
Zeros of Holomorphic Functions
Theorem: f(z) Analytic and Nonzero → ln|f(z)| Harmonic
Polynomial Theorem
Jensen’s Formula
Poisson-Jensen’s Formula for a Disk

6. 6
SOLO
Complex Variables
Table of Contents (continue - 4)
Calculation of the Residues
The Residue Theorem, Evaluations of Integral and Series
The Residue Theorem
Evaluation of Integrals
Jordan’s Lemma
Integral of the Type Bromwwich-Wagner
Integral of the Type ,F (sin θ, cos θ) is a
rational function of sin θ and cos θ
 



2
0
cos,sin dF
Definite Integrals of the Type . 



xdxF
Cauchy’s Principal Value
Differentiation Under Integral Sign, Leibnitz’s Rule
Summation of Series
Infinite Products
The Mittag-Leffler and Weierstrass , Hadamard Theorems
The Weierstrass Factorization Theorem
The Hadamard Factorization Theorem
Mittag-Leffler’s Expansion Theorem
Analytic Continuation
Conformal Mapping

7. 7
SOLO
Complex Variables
Douglas N. Arnold
Gamma Function
Bernoulli Numbers
Fourier Transform
Laplace Transform
Z Transform
Mellin Transform
Hilbert Transform
Zeta Function
Table of Contents (continue - 5)
Applications of Complex Analysis
References

8. 8
SOLO
Algebra
Set of Numbers – Examples
  xnumberrealaisxx ,:R Set of real numbers
  yxiyixznumbercomplexaiszzC ,,1,,:
Set of complex numbers
  ,3,2,1,0,1,2,3,,:  integeranisiiZ
Set of integers
 ,3,2,1,0,0: integernaturalaisnnN 
Set of positive integers
or natural numbers
 0,,,/:  qZqpwhereqprrQ Set of rational numbers
We have:
CZN  R
 QxxIR  R: Set of irrational numbers
 IRQIRQ  &R

9. 9
SOLO
Complex Variables
Complex numbers can result by solving algebraic equations
a
cabb
x
2
4
2
1


a
cabb
x
2
4
2
2


04
2
 cab
a
b
xx
2
21


04
2
 cab
04
2
 caby
x
cxbxay 
2
a
bcaib
x
2
4
2
2,1


y
x
dxcxbxay 
23
Three real roots for y = 0
One real & two complex
roots for y = 0
ki
ez
25
1
y
x
5
2
2

i
ez
5
2
2
2

i
ez
5
2
3
3

i
ez
5
2
4
4

i
ez
1
1
z

72

72

72

72

72
1. Quadratic equations
2. Cubic equations
3. Equation
Examples
0
2
 cxbxa
0
23
 dxcxbxa
01
5
x
Return to Table of Contents

10. 10
SOLO
Complex Variables
Fundamentals Operations with Complex Numbers z = x + i y
 
1
,sincos
,
2






 i
ArgumentModulusi
partImaginaryypartRealxyix
z



i
eyixz 
y
x


Division
Addition   dbicadicbia 
Subtraction   dbicadicbia 
Multiplication     cbdaidbcadbicbidaicadicbia 
2
 
 
 
 
 
 
  
  dccdidc
dacbidbca
dic
dic
dic
bia
dic
bia










22
 
 
 
 
 
 
0
22
2222









dc
dc
dacb
i
dc
dbca
dic
bia
Conjugate
  sincos:
*
iyixz 
Absolute Value

*22
: zzyxz


i
eyixz

:
*


i
eyixz :
x
y





11. 11
SOLO
Complex Variables
Fundamentals Operations with Complex Numbers z = x + i y


i
eyixz 
y
x


Polar Form of a Complex Number
Multiplication
Division
 
2121
212121




iii
eeezz
*22
zzyxz 
  sincos: iyixz 
xy/tan
1

 
2121
212121
///




iii
eeezz
Euler’s Formula
 



 
 




sincos
!12
1
!3!1!2
1
!4!2
1
!!2!1
1
sin
123
cos
242
2
i
k
i
k
n
iii
e
k
k
k
k
n
i























Leonhard Euler
1707- 1783
 
1
,sincos
,
2






 i
ArgumentModulusi
partImaginaryypartRealxyix
z


12. 12
SOLO
Complex Variables
Fundamentals Operations with Complex Numbers z = x + i y
1
,
,
2






 i
ArgumentModuluse
partImaginaryypartRealxyix
z
i




i
eyixz 
y
x


Polar Form of a Complex Number


i
eyixz :
De Moivre Theorem
  
 



nine
eiz
nnin
n
inn
sincos
sincos


Roots of a Complex Number
 
    
1.2.1.0
2
sin
2
cos
sincos
/1
/1/1
2/1





















nk
n
k
i
n
k
iez
n
nn
kin





ki
ez
25
1
y
x
5
2
2

i
ez
5
2
2
2

i
ez
5
2
3
3

i
ez
5
2
4
4

i
ez
1
1
z

72

72

72

72

72
Abraham De Moivre
1667 - 1754
Return to Table of Contents

13. 13
SOLO
Complex Variables
Axiomatic Foundations of the Complex Number System  R babaz ,,
Definition of Complex System:
From those relation, for any complex numbers z
1
,z
2
,z
3
 C we obtain:
CzzCzzCzz 
212121 ,& Closure Law 1
1221 zzzz  Commutative Law of Addition2
  
312321 zzzzzz  Associative Law of Addition3
1221 zzzz  Commutative Law of Multiplication4
  
312321
zzzzzz  Associative Law of Multiplication5
 
3121321 zzzzzzz  Distributive Law
6
111
00 zzz 
111
11 zzz 7
0..
11  zztsCzuniqueCz zz
18
1..0
11  zztsCzuniqueCz zzz /1
1
1


9
Equality  dbcadcba  ,,,A
Sum dbcadcba ,,
B
Product  cbdadbcadcba  ,,,   R mbmambam &,,C
Return to Table of Contents

14. 14
SOLO
Complex Variables
History of Complex Numbers
Brahmagupta (598-670) writes Khandakhadyaka
(665) which solves quadratic equations and allows
for the possibility of negative solutions.
Brahmagupta
598 - 670
Brahmagupta also solves quadratic equations of the type
a x
2
+ c = y
2
and a x
2
- c = y
2
. For example he solves 8x
2
+ 1 = y
2

obtaining the solutions (x,y) = (1,3), (6,17), (35,99), (204,577),
(1189,3363), ... For the equation 11x
2
+ 1 = y
2
Brahmagupta
obtained the solutions (x,y) = (3,10), (161/5,534/5), ... He also
solves 61x
2
+ 1 = y
2
which is particularly elegant having x =
226153980, y = 1766319049 as its smallest solution.

15. 15
SOLO
Complex Variables
History of Complex Numbers
Abraham bar Hiyya Ha-Nasi אישנה אייח רב םהרבא
writes the work Hibbur ha-Meshihah ve-ha-Tishboret
תרובשתהו החישמה רובח , translated in 1145 into Latin as Liber
embadorum, which presents the first complete solution to the
quadratic equation.
Abraham bar Hiyya Ha-Nasi (אישנה אייח רב םהרבא Abraham son of
[Rabbi] Hiyya "the Prince") (1070 - 1136?) was a Spaish Jewish
Mathematician and astronomer, also known as Savasorda (from the
Arabic
ةطرشلابحاص
Sâhib ash-Shurta "Chief of the Guard"). He lived
in Barcelona.
Abraham bar
Ḥiyya ha-Nasi
[2]

(1070 – 1136 or 1145)

16. 16
SOLO
Complex Variables
History of Complex Numbers
Nicolas Chuquet (1445 – 1488)
Chuquet wrote an important text Triparty en la science des nombres.
This is the earliest French algebra book .
The Triparty en la science des nombres (1484) covers arithmetic and
algebra. It was not printed however until 1880 so was of little
influence. The first part deals with arithmetic and includes work on
fractions, progressions, perfect numbers, proportion etc. In this work
negative numbers, used as coefficients, exponents and solutions,
appear for the first time. Zero is used and his rules for arithmetical
operations includes zero and negative numbers. He also uses x0 = 1
for any number x.
The sections on equations cover quadratic equations where he
discusses two solutions.

17. 17
SOLO
Complex Variables
History of Complex Numbers
Girolamo Cardano
1501 - 1576
Nicolo Fontana Tartaglia
1500 - 1557
Solution of cubic equation x
3
+ b x
2
+c x +d = 0
The first person to solve the cubic equation x
3
+b x = c was
Scipione del Ferro (1465 – 1526), but he told the solution only
to few people, including his student Antonio Maria Fior.
Nicolo Fontana Tartaglia, prompted by the rumors, manage
to solve the cubic equation x
3
+b x
2
= -d and made no secret of
his discovery.
Fior challenged Tartaglia, in 1535, to a public contest, each one
had to solve 30 problems proposed by the other in 40 to 50 days.
Tartaglia managed to solve his problems of type x
3
+m x = n in
about two hours, and won the contest.
News of Tartaglia victory reached Girolamo Cardan in Milan,
where he was preparing to publish Practica Arithmeticae (1539).
Cardan invited Tartaglia to visid him and, after much persuasion,
made him to divulge his solution of the cubic equation. Tartaglia
made Cartan promise to keep the secret until Tartaglia had
published it himself.

18. 18
SOLO
Complex Variables
History of Complex Numbers
Girolamo Cardano
1501 - 1576
Nicolo Fontana Tartaglia
1500 - 1557
Solution of cubic equation x
3
+ b x
2
+c x +d = 0
After Tartaglia showed Cardan how to solve cubic equations,
Cartan encouraged his student Lodovico Ferrari (1522 – 1565)
to use those result and solve quartic equations x
4
+p x
2
+q x +r=0.
Since Tartaglia didn’t publish his results and after hearing from
Hannibal Della Nave that Scipione del Ferro first solve cubic
equations, Cardan pubished in 1545 in Ars Magna (The Great Art)
the solutions of the cubic (credit given to Tartaglia) and quartic
equations.
This led to another competition between Tartaglia and
Cardano, for which the latter did not show up but was
represented by his student Lodovico Ferrari.
Ferrari did better than Tartaglia in the competition, and
Tartaglia lost both his prestige and income.

19. 19
SOLO
Complex Variables
History of Complex Numbers
Solution of cubic equation x
3
+ b x
2
+c x +d = 0
0
23
 dxcxbx
3/bxy

0
27
2
33
393
2
273333
33
2
3
3
22
32
23
23




































nymyb
cb
dy
b
cy
d
cb
yc
b
ybyb
b
y
b
ybyd
b
yc
b
yb
b
y
nm

Equivalence between x
3
+ b x
2
+c x +d = 0 and y
3
+m y = n (depressed cubic equation)
Solutions of y
3
+m y = n
Start from the identity: 

 nymybabababa
n
ymy

3333
3

0
27273
3
3
36
3
3
333

m
ana
a
m
aban
a
m
bbam
3
32
3,2,13,2,1
2742
mnn
wa 






























3
3
2
3
3
2
3,2,13,2,1
342
3
3423
mnn
m
mnn
w
a
m
ay
2
31
,
2
31
,1
3
3,2,1
ii
ew
ki
k





20. 20
SOLO
Complex Variables
History of Complex Numbers
Solution of cubic equation x
3
+ b x
2
+c x +d = 0





































































































3
3
2
3
3
2
3,2,1
3
322
3
3
2
3
3
2
3,2,1
3
3
2
3
3
2
3,2,13,2,1
342342
322
342
3342
342
3
3423
mnnmnn
w
mnn
mnn
mmnn
w
mnn
m
mnn
w
a
m
ay


2
31
,
2
31
,1
342342
3
3,2,1
3
3
2
3
3
2
3,2,13,2,1
ii
ew
mnnmnn
wy
ki
k



























Solutions of y
3
+m y = n
Note:
Tartaglia and
Cardano knew
only the solution
w
1
= 1

21. 21
SOLO
Complex Variables
History of Complex Numbers
Solution of cubic equation x
3
+ b x
2
+c x +d = 0
A Solution of y
3
+m y = n
Guess a solution: 






33
huhuwy
  uzhuhuhuhuhuhuy 233
3 2333 23







Therefore:





























32
3 2
32
2
3
2
mn
h
n
u
mhu
nu



































3
32
3
32
3,2,13,2,1
322322
mnnmnn
wy
Where w are the roots of w
3
=1:
2
31
,
2
31
,1
3,2,1
ii
w



22. 22
SOLO
Complex Variables
History of Complex Numbers
Solution of cubic equation x
3
+ b x
2
+c x +d = 0
Viète Solution of y
3
+m y = n
François Viète
1540 - 1603
In 1591 François Viète gave another solution to y
3
+m y = n
Start with identity CC
C
34cos3cos43cos
3
cos
3



Substitute in y
3
+m y = n C
m
y
3
2
3
3
3
2
1
3
3
3
2
34
3
2
3
8
3




















m
n
CCnC
m
mC
m
m
Assuming we obtain
3
23
32
1
3
2



















 mn
m
n


cos
3
2
3cos
23
3
k
m
n






























3
1 3
2
cos
3
2
cos
3
2
3
2
m
nkm
C
m
y 


23. 23
SOLO
Complex Variables
History of Complex Numbers
Solution of cubic equation x
3
+ b x
2
+c x +d = 0
Comparison of Cardano and Viète Solution of y
3
+m y = n
François Viète
1540 - 1603Cardano solution was






















3
1 3
2
cos
3
2
cos
3
2
m
nkm
ssy 

Girolamo Cardano
1501 - 1576
3
3
2
3
3
2
342342













mnnmnn
y
 















































3
1
2
3
3
2
3
3
2
cos
3
342
33
m
n
e
m
mnn
s
ssy
ki
ss



or
from which we recover Viète Solution
0s

2
n
3
s
3
s
0
x
3/
1s
2
s
0s
2s
1s
23
23











 nm
3
z

120

120

120

24. 24
SOLO
Complex Variables
History of Complex Numbers
Rafael Bombelli
1526 - 1572
John Wallis
1616 - 1703
In 1572 Rafael Bombelli published three of the intended five
volumes of “L’Algebra” worked with non-real solutions of the
quadratic equation x
2
+b x+c=0 by using and
where and applying addition and multiplication rules.
vu 1 vu 1
11
2

In 1673 John Wallis presented a geometric picture of the complex
numbers resulting from the equation x
2
+ b x + c=0, that is close
with what we sed today.
Wallis's method has the undesirable consequence that
      is represented by the same
point as is
   1
1
0,b0,b
bb
c
bb
c
1
P
2P
2P
1P
Wallis representation of real roots of
quadratics
Wallis representation of non-real roots of
quadratics

25. 25
Vector Analysis History SOLO
Caspar Wessel
1745-1818
“On the Analytic Representation
of Direction; an Attempt”, 1799
bia
Jean Robert Argand
1768-1822
1806
1i
3 .R.S. Elliott, “Electromagnetics”,pp.564-568
http://www-groups.dcs.st-and.ac.uk/~history/index.html
Wessel's fame as a mathematician rests solely on
this paper, which was published in 1799, giving for
the first time a geometrical interpretation of
complex numbers. Today we call this geometric
interpretation the Argand diagram but Wessel's
work came first. It was rediscovered by Argandin
1806 and again by Gauss in 1831. (It is worth
noting that Gauss redid another part of Wessel's
work, for he retriangulated Oldenburg in around
1824.)

26. 26
SOLO
Complex Variables
History of Complex Numbers
Leonhard Euler
1707- 1783
In 1748 Euler published “Introductio in Analysin Infinitorum” in
which he introduced the notation and gave the formula1i
xixe
ix
sincos
In 1751 Euler published his full theory of logarithms and
complex numbers. Euler discovered the Cauchy-Riemann
equations in 1777
although d’Alembert had discovered them in 1752 while
investigating hydrodynamics.
Johann Karl Friederich Gauss published the first correct proof
of the fundamental theorem of algebra in his doctoral thesis of
1797, but still claimed that "the true metaphysics of the square
root of -1 is elusive" as late as 1825. By 1831 Gauss overcame
some of his uncertainty about complex numbers and published
his work on the geometric representation of complex numbers as
points in the plane.
Karl Friederich Gauss
1777-1855

27. 27
SOLO
Complex Variables
History of Complex Numbers
Augustin Louis Cauchy
(

1789-1857)
Cauchy is considered the founder of complex analysis after
publishing the Cauchy-Riemann equations in 1814 in his paper
“Sur les Intégrales Définies”. He created the Residue Theorem and
used it to derive a whole host of most interesting series and integral
formulas and was the first to define complex numbers as pairs of
real numbers.
Georg Friedrich Bernhard
Riemann
1826 - 1866
In 1851 Riemann give a dissertation in the theory of functions.
Return to Table of Contents

28. 28
SOLO
Complex Variables
Derivatives
If f (z) is a single-valued in some region C of the z plane, the derivative of f (z) is
defined as:

 
z
zfzzf
zf
z 


0
lim'
provided that the limit exists independent of the manner in which Δ z→0.
In such case we say that f (z) is differentiable at z.
Analytic Functions
If the derivative of f (z) exists at all points of a region C of the z plane, then f (z) is
said to be analytic in C.
analytic = regular = holomorphic
A function f (z) is said to be analytic at a point z
0 if there exists a neighborhood
z-z
0
 < δ in which f ’ (z) exists.
z
0
δ
Analytic functions have derivatives of any order
which themselves are analytic functions.
Return to Table of Contents

29. 29
SOLO
Complex Variables
Derivatives
If f (z) is a single-valued in some region C of the z plane, the derivative of f (z) is
defined as:

 
z
zfzzf
zf
z



0
lim'
provided that the limit exists independent of the manner in which Δ z→0.
In such case we say that f (z) is differentiable at z.
Analytic Functions
If the derivative of f (z) exists at all points of a region C of the z plane, then f (z) is
said to be analytic in C.
analytic = regular = holomorphic
A function f (z) is said to be analytic at a point z
0 if there exists a neighborhood
z-z
0
 < δ in which f ’ (z) exists.
z
0
δ
Analytic functions have derivatives of any order
which themselves are analytic functions.

30. 30
SOLO
Complex Variables
Analytic, Holomorphic, MeromorphicFunctions
Return to Table of Contents
A Meromorphic Function on an open subset D of the complex plane is a
function that is Holomorphic on all D except a set of isolated points, which are
poles for the function. (The terminology comes from the Ancient Greek meros
(μέρος), meaning “part”, as opposed to holos ( λος)
ὅ
, meaning “whole”.)
The word “Holomorphic" was introduced by two of Cauchy's students, Briot
(1817–1882) and Bouquet (1819–1895), and derives from the Greek λος (holos)
ὅ
meaning "entire", and μορφή (morphē) meaning "form" or "appearance".
[2]
Today, the term "holomorphic function" is sometimes preferred to "analytic
function", as the latter is a more general concept. This is also because an
important result in complex analysis is that every holomorphic function is
complex analytic, a fact that does not follow directly from the definitions. The
term "analytic" is however also in wide use.
The Gamma Function is Meromorphic in the
whole complex plane
Poles

31. 31
SOLO
Complex Variables
Cauchy-Riemann Equations
A necessary (but not sufficient) condition that f (z) = u (x,y) +i v (x,y) be
analytic in a region C, is that u and v satisfy the Cauchy-Riemann equations:
y
u
x
v
y
v
x
u










&
Proof:
Augustin Louis Cauchy
(

1789-1857)
Georg Friedrich Bernhard
Riemann
1826 - 1866

 
z
zfzzf
zf
z



0
lim'
Provided that the limit exists independent of the manner in which
Δ z→0.
Choose Δ z = Δ x →

x
v
i
x
u
zf





'
Choose Δ z =i Δ y →

y
v
y
u
izf





'
Equalizing those two expressions we obtain:

y
u
i
y
v
x
v
i
x
u
zf











'
y
u
x
v
y
v
x
u










 &
The functions u (x,y) and v (x,y) are called conjugate functions,
because if one is given we can find the other (with an arbitrary
additive constant). Return to Table of Contents

32. 32
SOLO
Complex Variables
Harmonic Functions
If the second partial derivatives of u (x,y) and v (x,y) with respect to x and y exist
and are continuous in a region C, then using Cauchy-Riemann equations, we obtain:
0
2
2
2
2
2
22
2
2
2
22















































y
u
x
u
y
u
xy
v
y
u
x
v
yx
v
x
u
y
v
x
u
xyyx
y
x
0
2
2
2
2
2
2
2
2
22
22















































y
v
x
v
yx
u
x
v
y
u
x
v
y
v
xy
u
y
v
x
u
xyyx
x
y
It follows that under those conditions the real and imaginary parts
of an analytic function satisfy Laplace’s Equation denoted by:
0
2
2
2
2






yx
or 2
2
2
2
2
:
yxx
i
xx
i
x 

































where
0
2

The functions satisfying Laplace’s Equation are called Harmonic Functions.
Pierre-Simon Laplace
(1749-1827)
Return to Table of Contents

33. 33
SOLO
Complex Variables
Singular Points
A point at which f (z) is not analytic is called a singular point. There are various types
of singular points:
1. Isolated Singularity
The point z
0
at which f (z) is not analytic is called an isolated singular point, if we can
a neighborhood of z
0
in which there are not singular points.
z
0
δ
If no such a neighborhood of z
0
can be found then we
call z
0
a non-isolated singular point.
2. Poles
Example 
  
5353
1834
32



zzzz
zz
zf
has a pole of order 2 at z = 3, a pole of order 3 at z = 5, and two simple
poles at z = -3 and z = -5.
If we can find a positive integer n such that
and is analytic at z=z
0
then z = z
0
is called a pole of order n. If n = 1, z is called a simple pole.
  0lim
0
0


Azfzz
n
zz
 zfzzz
n
0


34. 34
SOLO
Complex Variables
Singular Points
A point at which f (z) is not analytic is called a singular point. There are various types
of singular points:
3. Branch Points
If f (z) is a multiple valued function at z
0, then this is a branch point.
Examples:
 
n
zzzf
/1
0
 has a branch point at z=z
0
   
0201ln zzzzzf  has a branch points at z=z
01
and z=z
02
4. Removable Singularities
The singular point z
0
is a removable singularity of f (z) if exists.zf
zz
0
lim

Examples: The singular point z = 0 of is a removable singularity
z
zsin
1
sin
lim
0


z
z
z

35. 35
SOLO
Complex Variables
Singular Points
A point at which f (z) is not analytic is called a singular point. There are various types
of singular points:
5. Essential Singularities
A singularity which is not a pole, branch point or a removable singularity is called
an essential singularity.
Example: has an essential singularity at z = z
0
.
 
0/1 zz
ezf


6. Singularities at Infinity
If we say that f (z) has singularities at z →∞. The type of the
singularity is the same as that of f (1/w) at w = 0.
0lim 

zf
z
Example: The function f (z) = z
5
has a pole of order 5 at z = ∞, since f (1/w) = 1/w
5
has a pole of order 5 at w = 0.
Return to Table of Contents

36. 36
SOLO
Complex Variables
Orthogonal Families
If f (z) = u (x,y) + i v (x,y) is analytic, then the one-parameter families of curves
   yxvyxu ,,,
where α and β are constant are orthogonal.
Proof:
The normal to u (x,y) = α is: y
y
u
x
x
u
yxu 11,






The normal to v (x,y) = β is: y
y
v
x
x
v
yxv 11,






The scalar product between the normal to u (x,y) = α and the normal to v (x,y) = β is:

y
v
y
u
x
v
x
u
yxvyxu









 ,,
Using the Cauchy-Riemann Equation for the analytic f (z):
y
u
x
v
y
v
x
u










&
 0,, 










y
v
y
u
y
u
y
v
yxvyxu
x
y
yxu,
yxv,
planez
u
v
planew
Return to Table of Contents

37. 37
SOLO
Complex Variables
Complex Line Integrals
Let f (z) be continuous at all points on a curve C of a finite length L.
  

 
n
i
ii
n
i
iiin zfzzfS
11
1 
C
1
z
nzb
2
z
0
za
1iz
i
z
1

2

i
n

Let subdivide C into n parts by n arbitrary points
z
1, z
2,…,z
n, and call a=z
0 and b=z
n. On each arc joining
z
i-1 to zi choose a point ξ
i. Define the sum:
Let the number of subdivisions n increase in such a
way that the largest of Δzi approaches zero, then the sum approaches a limit
that is called the line integral (also Riemann-Stieltjes integral).
  




C
b
a
n
i
ii
z
n
n
zdzfzdzfzfS
i
1
0
limlim 
Properties of Integrals
   


CCC
zdzgzdzfzdzgzf
  constantAzdzfAzdzfA
CC


 


a
b
b
a
zdzfzdzf
  


b
c
c
a
b
a
zdzfzdzfzdzf
   CoflengthLandConMzfLMzdzfzdzf
CC


Return to Table of Contents

38. 38
SOLO
Complex Variables
Simply and Multiply Connected Regions
A region R is called simply-connected if any simple closed curve Γ, which lies in R
can be shrunk to a point without leaving R. A region R that is not simply-connected
is called multiply-connected.
C
0
x
y
R
C
1

C
0
x
y
R
C
1
C
2
C
3

C
x
y
R

C
x
y
R
simply-connected
multiply-connected.
Return to Table of Contents

39. 39
SOLO
Complex Variables
Green’s Theorem in the Plane
C
R
Let P (x,y) and Q (x,y) be continuous and have continuous
partial derivatives in a region R and on the boundary C.
Green’s Theorem states that:
GEORGE STOCKES
1819-1903
A more general theorem was given by Stokes
 
 













R
dydx
y
P
x
Q
dyQdxP
C
 
 









































yzxzxy
RRR
dzdy
z
Q
y
R
dzdx
x
R
z
P
dydx
y
P
x
Q
dzRdyQdxP
C
or in vector form: 

S
dAFdrF
C

where:
      zzyxRyzyxQxzyxPzyxF 1,,1,,1,,,, 

zdzydyxdxdr 111 
zdydxydzdxxdzdydA 111 
GEORGE GREEN
1793-1841
z
z
y
y
x
x
111










40. 40
SOLO
Complex Variables
Proof of Green’s Theorem in the Plane C
R
P
T
S
Q
a b
x
y
xgy
2
xgy
1

Start with a region R and the boundary curve C, defined
by S,Q,P,T, where QP and TS are parallel with y axis.










b
a
xgy
Xgy
dy
y
P
dxdydx
y
P
2
R
By the fundamental lemma
of integral calculus:






   xgxPxgxPyxPdy
y
yxP xgy
xgy
xgy
Xgy
12
,,,
,
2
1
2


 




Therefore:    




b
a
b
a
dxxgxPdxxgxPdydx
y
P
12 ,,
R
but:
   


a
bSQ
dxxgxPdxxgxP
22
,,
integral along curve SQ
   


b
aPT
dxxgxPdxxgxP
11 ,, integral along curve PT
If we add to those integrals:   00,, 

dxsincedxyxPdxyxP
QPTS
we
obtain:
      




CTSPTQPSQ
dxyxPdxyxPdxxgxPdxyxPdxxgxPdydx
y
P
,,,,,
12
R
Assume that PT is defined by the function y = g
1 (x) and
SQ is defined by the function y = g
2 (x), both smooth and
y
P


is continuous in R:

41. 41
SOLO
Complex Variables
Proof of Green’s Theorem in the Plane (continue – 1)
In the same way:
Therefore we obtain:





C
dxyxPdydx
y
P
,
R





C
dyyxQdydx
x
Q
,
R
 
 













R
dydx
y
P
x
Q
dyQdxP
C
The line integral is evaluated by traveling C counterclockwise.
For a general single connected region, as that
described in Figure to the right, can be divided in a
finite number of sub-regions R
i, each of each are of
the type described in the Figure above. Since the
adjacent regions boundaries are traveled in opposite
directions, there sum is zero, and we obtain again:
 
 













R
dydx
y
P
x
Q
dyQdxP
C
C
R
4
x
y
R
R
3
R
1
R
2
C
R
P
T
S
Q
a b
x
y
xgy
2
xgy
1


42. 42
SOLO
Complex Variables
Proof of Green’s Theorem in the Plane (continue – 2)
The general multiply-connected regions can be transformed in a simply
connected region by infinitesimal slits
Since the slits boundaries are traveled in opposite
directions, there integral sum is zero:
C
0
x
y
R
C
1
P
0
P
1
C
0
x
y
R
C
1
C
2
C
3
   
 













R
dydx
y
P
x
Q
dyQdxPdyQdxP
i
CC
i0
All line integrals are evaluated by traveling C
i
i=0,1,… counterclockwise.
  0
0
1
1
0


P
P
P
P
dyQdxPdyQdxP
We obtain:
Return to Table of Contents

43. 43
SOLO
Complex Variables
Consequences of Green’s Theorem in the Plane
Let P (x,y) and Q (x,y) be continuous and have continuous first partial
derivative at each point of a simply-connected region R. A necessary and
sufficient condition that around every closed path C in R
is that in R. This is synonym to the condition that
is path independent.
y
P
x
Q





 0

C
dyQdxP
Sufficiency:
Suppose
y
P
x
Q





According to Green’s Theorem  0















R
dydx
y
P
x
Q
dyQdxP
C
Necessity:
0





or
y
P
x
Q
Suppose along every path C in R. Assume that
at some point (x
0,y
0) in R. Since  Q/  x and  P/  y are continuous exists
a region τ around (x
0,y
0) and boundary Γ for which , therefore:
 0

C
dyQdxP
0





or
y
P
x
Q
  0
















ordydx
y
P
x
Q
dyQdxP

C
x
y
R
 


L
dyQdxP
0





y
P
x
Q
This is a contradiction to the assumption, thereforeq.e.d.
Return to Table of Contents

44. 44
SOLO
Complex Variables
Cauchy’s Theorem
C
x
y
R
Proof:
0

C
dzzf
If f (z) is analytic with derivative f ‘ (z) which is continuous at all points inside
and on a simple closed curve C, then:
yxviyxuzf ,, Since is analytic and has continuous
first order derivative

y
u
i
y
v
x
v
i
x
u
zd
fd
zf
iyzxz













'
y
u
x
v
y
v
x
u










& Cauchy - Riemann
      
0
00
































RR
dydx
y
v
x
u
idydx
y
u
x
v
dyudxvidyvdxudyidxviudzzf
CCCC

q.e.d.
Augustin Louis Cauchy
(

1789-1857)
Return to Table of Contents

45. 45
SOLO
Complex Variables
Cauchy-Goursat Theorem
C
x
y
R
Proof:
0

C
dzzf
If f (z) is analytic which is continuous at all points inside and on a simple
closed curve C, then:
Augustin Louis Cauchy
(

1789-1857)
Goursat removed the Cauchy’s condition
that f ‘ (z) should be continuous in R.
C
F
DE
A
B
I

IV
II
III

Start with a triangle ABC in z in which
f (z) is analytic, Join the midpoints E,D,F
to obtain four equal triangles Δ
I, Δ
II, Δ
III,
Δ
IV
. We have:

   




















































IVIIIIII
dzzfdzzfdzzfdzzf
dzzf
DEFDFCDFEBFEDAED
FDEFDEDFFCDFEEBFEDDAE
FCDEBFDAEABCA
Eduard Jean-Baptiste
Goursart
1858 - 1936

46. 46
SOLO
Complex Variables
Proof of Cauchy-Goursat Theorem (continue – 1)
If f (z) is analytic which is continuous at all points inside and on a simple
closed curve C, then:
C
F
DE
A
B
I
IV
II
III
    



IVIIIIII
dzzfdzzfdzzfdzzfdzzf
ABCA
then:
    



IVIIIIII
dzzfdzzfdzzfdzzfdzzf
ABCA
Let Δ
1
be the triangle in which the absolute value of the integral is maximum.
 



1
4 dzzfdzzf
Continue this procedure in triangle Δ
1 in which Δ
2 is the triangle in which the
absolute value of the integral is maximum.
  



21
2
44 dzzfdzzfdzzf
 



n
dzzfdzzf
n
4

47. 47
SOLO
Complex Variables
Proof of Cauchy-Goursat Theorem (continue – 2)
C
F
DE
A
B
I
IV
II
III

 



n
dzzfdzzf
n
4
For an analytic function f (z) compute


0
0
0
0
':, zf
zz
zfzf
zz 





 0'''lim,lim
000
0
0
0
00












zfzfzf
zz
zfzf
zz
zzzz

   
0000000 ,'&,..,0 zzzzzzzfzfzfzzwheneverzzts  
      





nnnn
dzzzzzdzzzzzdzzzzfzfdzzf
TheoremIntegralCauchy
0000
0
)00 ,,' 

n
0z
na
n
b
nc
z
0zz
0
zzcbaP
nnnn

  
2
2
00
2
, 








nnn
P
PdzPdzzzzzdzzf
nnn

But , where P
n
the perimeter
of Δ
n and P the perimeter of Δ are related, by construction, by
  
nPzzzz
00 &,
n
n
PP 2/
q.e.d.
  0
4
44
0
2
2





dzzfP
P
dzzfdzzf
n
nn
n



48. 48
SOLO
Complex Variables
Proof of Cauchy-Goursat Theorem (continue – 3)
n
z
1z
2
z
1iz
i
z
1n
z
n
1
2
3
i
C
O
q.e.d.
For the general case of a simple closed curve C
we take n points on C: z
1
, z
2
,…,z
n
and a point
O inside C. We obtain n triangles Δ
1, Δ
2,.., Δ
n,
for each of them we proved Cauchy-Goursat Theorem.
Let define the sum: 





n
i
zz
iin
ii
zzfS
1
1
:
we have:
   0
1
1




i
i
i
ii
z
O
O
z
z
z
dzzfdzzfdzzfdzzf
      

n
i
i
i
i
i
i
i
ii
S
n
i
z
z
i
n
i
z
z
i
n
i
z
z
ii
n
i
z
z
n
i
dzzfdzzfzfdzzfzfzfdzzfdzzf 




11111
1111
0
  

NnforSdzzfdzzfS
n
CC
n
n


2
lim
    
22
1
1
111
11

 




n
i
ii
n
i
i
n
i
z
z
in
n
i
z
z
in
zz
L
dzfzfdzzfzfSdzzfzfS
i
i
i
i
   0
22


C
nn
CC
dzzfNnforSSdzzfdzzf 

Since we proved that , we can write:
0


dzzf
Return to Table of Contents

49. 49
SOLO
Complex Variables
Consequences of Cauchy-Goursat Theorem
B
x
y
R
A
C
1
C
2
D
1
D
2
a
b
If f (z) is analytic in a simply-connected region R, then
is independent of the path in R joining any
two points a and b in R.


b
a
dzzf
Let look at thr closed path AC
1
D
1
BD
2
C
2
A in R inside which f (z) is analytic.
According to Cauchy-Goursat Theorem
      


BDAcBDAcBDAcBDAcACBDBDAcACBDDAC
dzzfdzzfdzzfdzzfdzzfdzzfdzzf
2211221122111122
0
Proof:
If f (z) is analytic in a multiply-connected region R,
bounded by two simple closed curves C
1
and C
2
, then:
1
2
C
1
x
y
R
C
2
P
0
P
1
 


21 CC
dzzfdzzf
The general multiply connected regions can be transformed
in a single connected region by an infinitesimal slit P
0
to P
1
.
     


212
0
1
1
01
0
0
CCC
P
P
P
PC
dzzfdzzfdzzfdzzfdzzfdzzf

Proof:
Return to Table of Contents

50. 50
SOLO
Complex Variables
Cauchy’s Integral Formulas
Augustin Louis Cauchy
(

1789-1857)
If f (z) is analytic inside and on a simple closed curve C
and a is any point inside C then





C
dz
az
zf
i
af
2
1
C
x
y
R
a

Proof:
Let chose a circle Γ with center at a
 

2,0,: 
i
eazz
Since f (z)/ (z-a) is analytic in the region defined
between C and the circle Γ we can use:
 





zd
az
zf
zd
az
zf
C
  
 afidafidei
e
eaf
zd
az
zf
i
i
i








21lim
2
0
2
0
0



  

therefore:





C
dz
az
zf
i
af
2
1
q.e.d.
Cauchy’s Integral Formulas and Related Theorems
Return to Table of Contents

51. 51
SOLO
Complex Variables
Cauchy’s Integral Formulas for the n Derivative of a Function
Augustin Louis Cauchy
(

1789-1857)
If f (z) is analytic inside and on a simple closed curve C
and a is any point inside C, where the n derivative exists, then







C
n
n
dz
az
zf
i
n
af
1
2
!

C
x
y
R
a

Proof:
Let prove this by induction.
Assume that this is true for n-1:
Then we can differentiate under the sign of integration:





C
dz
az
zf
i
af
2
1
For n = 0 we found


 





C
n
n
dz
az
zf
i
n
af
2
!1
1


















C
n
C
n
nn
dz
az
n
zf
i
n
dz
azad
d
zf
i
n
af
ad
d
af
1
1
2
!11
2
!1

q.e.d.
Therefore for n we obtain:







C
n
n
dz
az
zf
i
n
af
1
2
!

We can see that an analytic function has derivatives of all orders.
Return to Table of Contents

52. 52
SOLO
Complex Variables
Morera’s Theorem (the converse of Cauchy’s theorem)
If f (z) is continuous in a simply-connected region R and if
around every simple closed curve C in R then
f (z) ia analytic in R.
0

C
dzzf
B
x
y
R
A
C
1
C
2
D
1
D
2
a
z
Proof:
Since around every closed curve C in R
0

C
dzzf
      


BDAcBDAcBDAcBDAcACBDBDAcACBDDAC
dzzfdzzfdzzfdzzfdzzfdzzfdzzf
2211221122111122
0
The integral is independent on path
between two points, if the path is in R
 


z
a
dzzfzF
Let choose a straight path between z and z+Δz
 
     




















zz
z
z
a
zz
a
udzfuf
z
zfudufuduf
z
zf
z
zFzzF 11
Since f (z) is continuous    zuwheneverzfuf
Therefore
 
 

zf
zd
zFd
zudzfuf
z
zf
z
zFzzF
zz
z








1
C
x
y
R
z

z+ z
Since F (z) has a derivative in R, it is analytic, and so are its derivatives, i.e. f (z)
Return to Table of Contents
Giacinto Morera
1856 - 1907

53. 53
SOLO
Complex Variables
Cauchy’s Inequality
Augustin Louis Cauchy
(

1789-1857)
If f (z) is analytic inside and on a circle C of radius r and
center at z = a, then

 ,...2,1,0
!


 n
r
nM
af
n
n
where M is a constant such that | f (z) |< M is an upper bound
of | f (z) | on C.
Proof:
C
x
y
a
r
Use Cauchy Integral Formula:




,2,1,0
!
2
1
2
!1
2
!
2
!
1
2
0
11





 
n
r
nM
r
r
Mn
dr
r
Mn
dz
az
zfn
af
nnn
Mzf
C
n
n









,2,1,0
2
!
1


 
ndz
az
zf
i
n
af
C
n
n

q.e.d.
On the circle C: .
i
eraz
Return to Table of Contents

54. 54
SOLO
Complex Variables
Liouville’s Theorem
Joseph Liouville
1809 - 1882
If for all z in the complex plane:
(1) f (z) is analytic
(2) f (z) is bounded, i.e. | f (z) |< M for some constant M
then f (z) must be a constant.
Proof No. 1:
Using Cauchy’s Inequality:

 ,...2,1,0
!


 n
r
nM
af
n
n
Letting n=1 we obtain: 
r
M
af
'
Since f (z) is analytic in all z plane we can take r → ∞ to obtain
  constantafaaf
r
M
af
r


,00lim
''
q.e.d.

55. 55
SOLO
Complex Variables
Liouville’s Theorem
Joseph Liouville
1809 - 1882
If for all z in the complex plane:
(1) f (z) is analytic
(2) f (z) is bounded, i.e. | f (z) |< M for some constant M
then f (z) must be a constant.
Proof No. 2:
Using Cauchy’s Integral Formula
q.e.d.
Let a and b be any two points in z plane. Draw a circle
C with center at a and radius r > 2 | a-b |
C
x
y
a
r
b
2/rba

  








CCC
dz
bzaz
zf
i
ab
dz
az
zf
i
dz
bz
zf
i
afbf
 22
1
2
1
We have
raz  2/rbarbaazbaazbz 




 r
Mab
dr
rr
Mab
dz
bzaz
zfab
dz
bzaz
zfab
afbf
CC











2
2/222
2
0



Since f (z) is analytic in all z plane we can take r → ∞ to obtain
 afbfafbf  0 therefore f (z) is constant.
Return to Table of Contents

56. 56
SOLO
Complex Variables
Foundamental Theorem of Algebra
Every polynomial equation P (z) = a
0
+ a
1
z+a
2
z
2
+…+a
n
z
n
=0 with degree
n ≥ 1 and a
n
≠ 0 (a
i
are complex constants) has at least one root.
From this it follows that P (z) = 0 has exactly n roots, due attention being
paid to multiplicities of roots.
Proof:
If P (z) = 0 has no root, then f (z) = 1 / P (z) is analytic for all z.
Also | f (z) |= 1 / | P (z) | is bounded. Then by Liouville’s Theorem f (z) and
then P (z) are constant. This is a contradiction to the fact that P (z) is a
polynomial in z, therefore P (z) = 0 must have at least one root (zero).
Suppose that z = a is one root of P (z) = 0. Hence P (a) = 0 and
  
   zQazazaazaaza
aaaaaaazazazaaaPzP
nn
n
n
n
n
n




22
21
2
210
2
210
Since a
n
≠ 0, Q (s) is a polynomial of degree n-1.
Applying the same reasoning to the polynomial Q (s) of degree n-1,
we conclude that it must also have at least one root. This procedure continues
until n = 0, therefore it follows that P (z) has exactly n roots.
q.e.d.
Return to Table of Contents

57. 57
SOLO
Complex Variables
Gauss’ Mean Value Theorem
Karl Friederich Gauss
1777-1855
C
x
y
a
r
If f (z) is analytic inside and on a circle C with center at
a and radius r, then f (a) is the mean of the values of f (z)
on C, i.e.,
  .
2
1
2
9






derafaf
i
Proof:
Use Cauchy Integral Formula:
On the circle C:
 ii
eridzeraz  .





C
dz
az
zf
i
af
2
1

  
 





 






derafderi
er
eraf
i
dz
az
zf
i
af
ii
i
i
C
2
1
2
1
2
1
q.e.d.
Return to Table of Contents

58. 58
SOLO
Complex Variables
Maximum Modulus Theorem
If f (z) is analytic inside and on a simple closed curve C and is not identically
equal to a constant, then the maximum value of | f (z) | occurs on C.
Proof:
The proof is based on the continuity of f (z) and on the Gauss’ Mean Value Theorem.
C
x
y
a
r
C
C
1
C
2
C
3
R

b
Since f (z) is analytic in C, | f (z) | has a maximum M inside or
on C. Suppose that the maximum value is achieved at the point
a inside C, i.e. | f (a) |=M =max | f (z) |. Since a is inside C we can
find a circle C
1
, with center at a that is inside C. Since f (z) is not
constant we can find a point b in C
1 such that | f (b)|=M – ε< | f (a)|.
Using the continuity of f (z) we can find a circle around b, C
2
,
   bzforbfzfzC 2/:
2
 2/2/2/   MMbfzf
Now apply the Gauss’ Mean Value Theorem for point a and the circle with center at a
passing trough b, C
3. Define by α the arc of C
3 inside C
2.
  .
2
1
2
9






derafaf
i
          















4
2
22
2/.
2
1
2
9
2/
2
9



M
M
MderafderafderafMaf
M
i
M
ii


59. 59
SOLO
Complex Variables
Maximum Modulus Theorem
If f (z) is analytic inside and on a simple closed curve C and is not identically
equal to a constant, then the maximum value of | f (z) | occurs on C.
Proof (continue):
          















4
2
22
2/.
2
1
2
9
2/
2
9



M
M
MderafderafderafMaf
M
i
M
ii

We obtained that is impossible, therefore
a for which |f (z)| is maximum cannot be inside C, but on C.



4
?
 MMaf
C
x
y
a
r
C
C
1
C
2
C
3
R

b
q.e.d.
Return to Table of Contents

60. 60
SOLO
Complex Variables
Minimum Modulus Theorem
If f (z) is analytic inside and on a simple closed curve C and f (z) ≠ 0
inside C then | f (z) | assumes its minimum value on C.
Proof:
Since f (z) is analytic inside and on a simple closed curve C and f (z) ≠ 0
inside C it follows that 1/ f (z) is analytic inside and on C. Then according to
Maximum Modulus Theorem 1/| f (z) | assumes its maximum vale on C and
therefore | f (z) | assumes its minimum value on C.
q.e.d.
x
yC
R
Return to Table of Contents

61. 61
SOLO
Complex Variables
Poisson’s Integral Formulas for a Circle
Siméon Denis Poisson
1781-1840
Let f (z) be analytic inside and on the circle C defined by
|z| = R, and let z = r e
iθ
be any point inside C, then:
 


  










 












2
0
2
0
2
2
2
2
22
2
0
22
22
2
1
2
1
cos22
1
deRf
zeR
zR
deRf
ereR
rR
deRf
rrRR
rR
erzf
i
i
i
ii
ii
C
x
y
R
R
z'
r
R
2
/r


z
z
r


 zRz /
2
1
Proof:
Since f (z) is analytic in C we can apply Cauchy’s Integral
Formula:





C
i
dz
zz
zf
i
erfzf '
'
'
2
1







C
dz
zRz
zf
i
'
/'
'
2
1
0
2

If we subtract those equations we obtain:
The inverse of the point z with respect to C is and
lies outside C, therefore by Cauchy’s Theorem:

 zRz /
2
1
 
 
















CC
i
dzzf
zRzzz
zRz
i
dzzf
zRzzzi
erfzf ''
/''
/
2
1
''
/'
1
'
1
2
1
2
2
2



62. 62
SOLO
Complex Variables
Poisson’s Integral Formulas for a Circle
Siméon Denis Poisson
1781-1840
Proof (continue):

 


   

 

  

 
  

 















































2
0
22
22
2
0
22
2
0
22
2
0
2
2
2
2
cos22
1
2
1
2
1
/
/
2
1
''
/''
/
2
1
deRf
RrRr
rR
deRf
ereRereR
rR
deRf
eRerereR
eRr
deRieRf
erReRereR
erRer
i
dzzf
zRzzz
zRz
i
erfzf
i
i
iiii
i
iiii
i
ii
iiii
ii
C
i
Writing we have:

,, rviruerf
i


 









2
0
22
22
,
cos22
1
, dRu
RrRr
rR
ru

 









2
0
22
22
,
cos22
1
, dRv
RrRr
rR
rv
q.e.d.
C
x
y
R
R
z'
r
R
2
/r


z
z
r


 zRz /
2
1
Return to Table of Contents

63. 63
SOLO
Complex Variables
Poisson’s Integral Formulas for a Half Plane
Siméon Denis Poisson
1781-1840
C
x
y
R

z
z
R
Let f (z) be analytic in the upper half y ≥ 0 of the z plane
and let z = (x + i y) any point in this upper half plane, then:






 dw
yxw
wfy
zf
22
Proof:
Let C be the boundary of a a semicircle of radius R
containing as an interior point, but does
not contain
yixz
yixz 

Using Cauchy’s Integral Formula we have:





C
dw
zw
wf
i
zf
2
1
By subtraction we obtain:





C
dw
zw
wf
i2
1
0
 
 

  
  
























CC
CC
dwwf
yxw
yi
i
dwwf
yixwyixw
yixyix
i
dwwf
zwzw
zz
i
dwwf
zwzwi
zf
22
2
2
1
2
1
2
111
2
1



64. 64
SOLO
Complex Variables
Poisson’s Integral Formulas for a Half Plane
Siméon Denis Poisson
1781-1840
Proof (continue):
Where Γ is the upper a semicircle of radius R.


















dwwf
yxw
y
dwwf
yxw
y
dwwf
yxw
yi
i
zf
R
R
C
2222
22
11
2
2
1


If we take R→∞ we obtain:

 0lim
1
22




dwwf
yxw
y
R







 dw
yxw
wfy
zf
22
Therefore:






 dw
yxw
wufy
yxu
22
0,
,






 dw
yxw
wvfy
yxv
22
0,
,
Writing and since w varies on x axis
, and we have:
yxviyxuzf ,,
0,0, wviwuwf 
q.e.d.
C
x
y
R

z
z
R
Return to Table of Contents

65. 65
SOLO
Infinite Series
Given a series:
Theorems of Convergence of Sequences and Series



n
i
in uS
1Convergence Definition:
The series S
n
converges to S as n →∞ if for all ε > 0 there exists an positive integer N
such that
If no such N exists then we say that the series diverges.
NnallforSuSS
n
i
in 


1
Convergence Theorem:
The series S
n
converges as n →∞ if and only if there exists an positive integer M
such that
If no such M exists then we say that the series diverges.
1
1


NallforMuS
N
i
iN
If S is unknown we can use the Cauchy Criterion for convergence:
for all ε > 0 there exists an positive integer N such that
NmnallforuuSS
m
j
j
n
i
imn  

,
11

Augustin Louis Cauchy
(

1789-1857)
A necessary (but not sufficient) condition for convergence is that lim
i→∞
u
i
= 0
Return to the Table of Content
Cauchy Convergence Criterion

66. 66
SOLO
Infinite Series
Given a series:
Theorems of Convergence of Sequences and Series



n
i
in uS
1
Convergence Tests
Cauchy Root Kummer, an
Euler-Maclurin
Integral
D’Alembert
Cauchy Ratio
Raabe
Gauss
(Comparison with
Geometric Series)
(Also by Comparison with
Geometric Series)
(Comparison with
Integral)
1
na na
n

nna
nln
In term by term a series of terms 0 ≤ un ≤ an, in which the an form a convergent series,
then is also convergent.
n
n
u
Return to the Table of Content

67. 67
SOLO
The Geometric Series


 
 r
r
a
r
rrarararaa
r
rS
nn
nG










1
1
1
1
1
1
132
1 
Multiply and divide by (1 – r)
 



 
1
0
132
1
n
i
in
nG rararararaaS 
We can see that















diverger
converger
r
a
r
r
aS
n
n
nG
n
1
1
1
1
1
limlim
1
Given a series:
Theorems of Convergence of Sequences and Series



n
i
in uS
1
Infinite Series

68. 68
SOLO
Convergence Tests
Cauchy Root Test
Augustin Louis Cauchy
(

1789-1857)
If (a
n
)
1/n
≤ r < 1 for all sufficiently large n, with r
independent of n, then is convergent. If (a
n
)
1/n
≥ 1 for
all sufficiently large n, then is divergent.

n
na

n
n
a
The first part of this test is verified easily by raising
(a
n
)
1/n
≤ r to the n
th
power. We get:
1
n
nra

n
n
a Since r
n
is just the n
th
term in a Convergent Geometric Series, is
convergent by the Comparison Test. Conversely, if (a
n
)
1/n
≥ 1, the a
n
≥ 1 and the
series diverge. This Root Test is particularly useful in establishing the properties
of Power Series.
Given a series:
Theorems of Convergence of Sequences and Series



n
i
in uS
1
Infinite Series
Return to the Table of Content

69. 69
SOLO
Convergence Tests
D’Alembert or Cauchy Ratio Test
Jean Le Rond D’Alembert
1717 - 1783
If (a
n+1
/a
n
) ≤ r < 1 for all sufficiently large n, with r
independent of n, then is convergent. If (a
n+1
/a
n
) ≥ 1 for
all sufficiently large n, then is divergent.

n
na

n
na
Convergence is proved by direct comparison with the
geometric series (1+r+r
2
+
…
)










ateindetermin,1
,1
,1
lim
1
divergence
econvergenc
a
a
n
n
n



n
i
n
n
nS
1
2/Example: convergent
n
n
a
a
n
n
n
n
n
n 2
12
2
1
limlim
1
1







Given a series:
Theorems of Convergence of Sequences and Series



n
i
in uS
1
Infinite Series
Return to the Table of Content
Augustin Louis Cauchy
(

1789-1857)

70. 70
SOLO
Convergence Tests
Maclaurin or Euler Integral Test
Given a series:
Theorems of Convergence of Sequences and Series



n
i
in uS
1
Infinite Series
Return to the Table of Content
Let f (n) = a
n
, i.e. f (x) is a monotonic decreasing function.
Then converges if is finite and diverges if the
integral is infinite.

n
n
a 

1
xdxf
  1
111
fxdxfaxdxf
n
n  



Colin Maclaurin
1698 - 1746
Leonhard Euler
(1707 – 1`783)
Is geometrically obvious that:
xf
x
1234

1
1af

22af
Comparison of Integral and Sum-Blocks
Leading
xf
x12 34

11af
Comparison of Integral and Sum-Blocks
Lagging

71. SOLO
Convergence Tests
Kummer’s Test
Consider a Series of positive terms u
i
and a sequence of positive constants a
i
.
If
for all n ≥ N, where N is some fixed number, then converges.
If
and diverges, then diverges.
The two tests can be written as:
Given a series:
Theorems of Convergence of Sequences and Series



n
i
in uS
1
Infinite Series


1i
iu
0
1
1



Ca
u
u
a
n
n
n
n
0
1
1



n
n
n
n a
u
u
a


1i
iu



1
1
i
ia


















divergea
converge
Ca
u
u
a
i
n
n
n
n
n
11
1 &0
0
lim
Ernst Eduard Kummer
(1810 – 1893)

72. 72
SOLO
Convergence Tests
Kummer’s Test (continue – 1)
Consider a Series of positive terms u
i and a sequence of positive
constants a
i
.
If
for all n ≥ N, where N is some fixed number, then converges.
0
1
1



Ca
u
u
a
n
n
n
n


1i
i
u
Ernst Eduard Kummer
(1810 – 1893)
nnnnn
NNNNN
NNNNN
uauauC
uauauC
uauauC






11
22112
111

Proof:
Add and divide by C
C
ua
C
ua
u
nnNN
n
Ni
i 
1
C
ua
u
C
ua
C
ua
uuuuS
NN
N
i
i
nnNN
N
i
i
n
Ni
i
N
i
i
n
i
in
 
 11111
The partial sums S
n
have an upper bound. Since the lower bound is zero
the sum must converge.iu
q.e.d.
Given a series:
Theorems of Convergence of Sequences and Series



n
i
in uS
1
Infinite Series

73. 73
SOLO
Convergence Tests
Kummer’s Test (continue – 2)
Given a series:
Theorems of Convergence of Sequences and Series



n
i
in uS
1
Infinite Series
Consider a Series of positive terms u
i
and a sequence of positive
constants a
i
.
If
and diverges, then diverges.
0
1
1



n
n
n
n
a
u
u
a


1i
i
u



1
1
i
ia
Proof: Nnuauaua
NNnnnn 
 ,
11 
Since a
n
> 0
n
NN
n
a
ua
u
and 






1
1
1 Ni
iNN
Ni
i auau
If diverges, then by comparison test diverges.



1
1
i
i
a 

1i
i
u
Return to the Table of Content
Ernst Eduard Kummer
(1810 – 1893)

74. 74
SOLO
Convergence Tests
Raabe’s Test
If u
n
> 0 and if
for all n ≥ N, where N is a positive integer independent on, then converges.
If
Then diverges (as diverges). The limit form of Raabe’s test is
Proof:
In Kummer’s Test choose a
n
= n and P = C + 1.
Joseph Ludwig Raabe
(1801 – 1859)
Given a series:
Theorems of Convergence of Sequences and Series



n
i
in uS
1
Infinite Series
Return to the Table of Content
11
1











P
u
u
n
n
n

i
iu
11
1










n
n
u
u
n

i
iu 

i
i
a
1

















 



testno
divergea
converge
P
u
u
n
i
i
n
n
n
1
&1
1
1lim
1
1

75. 75
SOLO
Convergence Tests
Gauss’ s Test
Carl Friedrich Gauss
(1777 – 1855)
If u
n > 0 for all finite n and
in which B (n) is a bounded function of n for n → ∞, then
converges for h > 1 and diverges for h ≤ 1. There is no
indeterminate case here.

2
1
1
n
nB
n
h
u
u
n
n



n
nu
Proof:
For h > 1 and h < 1 the proof follows directly from Raabe’s Test:
 
h
n
nB
h
n
nB
n
h
n
u
u
n
nn
n
n
n



























lim11lim1lim
2
1
If h = 1, Raabe’s Test fails. However if we return to Kummer’s Test and use a
n
=n ln
n:


 







































n
nnnn
n
n
nn
nn
n
nB
n
h
nn
nn
h
n
1
1lnlnln1lim1ln1
1
lnlim
1ln11lnlim
1
2
Given a series:
Theorems of Convergence of Sequences and Series



n
i
in uS
1
Infinite Series

76. 76
SOLO
Convergence Tests
Gauss’ s Test
Carl Friedrich Gauss
(1777 – 1855)
If u
n > 0 for all finite n and
in which B (n) is a bounded function of n for n → ∞, then
converges for h > 1 and diverges for h ≤ 1. There is no
indeterminate case here.

2
1
1
n
nB
n
h
u
u
n
n



n
nu
Proof (continue – 1):
Kummer’s witha
n=n ln n:

 





























 n
nnn
n
nB
n
h
nn
n
h
n
1
1ln1lim1ln11lnlim
1
2
  01
3
1
2
11
1lim
1
1ln1lim
32


















nnn
n
n
n
nn
Hence we have a divergence for h = 1. This is an example of a successful
application of Kremmer’s Test in which Raabe’s Test failed.
Given a series:
Theorems of Convergence of Sequences and Series



n
i
in uS
1
Infinite Series
Return to the Table of Content

77. 77
SOLO
Complex Variables
Infinite Series, Taylor’s and Laurent Series
Let {u
n
} :=u
1
(z), u
2
(z),…,u
n
(z),…, be a sequence of single-valued functions of z in
some region of z plane.
We call U (z) the limit of {u
n} ,if given any positive number ε we can find a number
N (ε,z) such that and we write this: zNnzUzu
n , 
  zUzuorzUzu
n
nn
n


lim
x
yC
R
If a sequence converges for all values z in a region R, we call R
the region of convergence of the sequence. A sequence that is
not
convergent at some point z is called divergent at z.
Infinite Series of Functions

78. 78
SOLO
Complex Variables
Infinite Series, Taylor’s and Laurent Series
Infinite Series of Functions
From the sequence of functions {u
n
} let form a new sequence {S
n
} defined by:


  




n
i
inn zuzuzuzuzS
zuzuzS
zuzS
1
21
212
11


If , the series is called convergent and S (z) is its sum.
zSzS
n
n


lim
A necessary (but not sufficient) condition for convergence is that lim
n→∞
u
n
(z) = 0
Example: The Harmonic Series
 

 nn
n
1
4
1
3
1
2
1
1
1
1
0
1
limlim 
 n
u
n
n
n
By grouping the terms in the sum as
































2
1
22
1
2
1
1
2
1
1
1
8
1
7
1
6
1
5
1
4
1
3
1
2
1
1
p
p
pppp
Return to the Table of Content

79. 79
SOLO
Complex Variables
Absolute Convergence of Series of Functions
Given a series of functions:
 


n
i
in zuzS
1
If is convergent the series is called absolutely convergent.

n
i
i
zu
1
If is convergent but is not, the series is called

conditionally convergent.


n
i
izu
1


n
i
izu
1
Return to the Table of Content

80. 80
SOLO
Complex Variables
Uniformly Convergence of Sequences and Series
If for the sequence of functions {u
n
(z)} we can find for each ε>0 a number N (ε)
such that for all zR we say that {u
n} uniformly
converges to U (z). ( N is a function only of ε and not of z)
  NnzUzu
n 
If the series of functions {S
n
(z)} converges to S (z) for all zR
we define the remainder 



1
:
nz
inn
zuzSzSzR
The series of functions {S
n
(z)} is uniformly convergent to S (z)
if for all for all ε>0 and for all zR we can find a number N (ε)
such that
  NnzSzS
n 
x
yC
R
Return to the Table of Content

81. 81
SOLO
Complex Variables
Weierstrass M (Majorant) Test
Karl Theodor Wilhelm
Weierstrass
(1815 – 11897)
The most commonly encountered test for Uniform
Convergence is the Weierstrass M Test.
Proof:
Since converges, some number N exists such that for n + 1 ≥ N,
If we can construct a series of numbers , in
which M
i
≥ |u
i
(x)| for all x in the interval [a,b] and
is convergent, the series u
i
(x) will be uniformly
convergent in [a,b].


1
iM


1
i
M


1
iM


1ni
iM
This follows from our definition of convergence. Then, with |u
i
(x)| ≤ M
i
for
all x in the interval a ≤ x ≤ b,

1ni
i
xu
Hence   

1ni
in xuxsxS
and by definition is uniformly convergent in [a,b]. Since we
specified absolute values in the statement of the Weierstrass M Test, the
series is also Absolutely Convergent.Return to the Table of Content


1i
iu


1i
iu

82. SOLO
Complex Variables
Abel’s Test
Niels Henrik Abel
( 1802 – 1829)
If
and the functions f
n
(x) are monotonic decreasing |
f
n+1
(x) ≤ f
n
(x)| and bounded, 0 ≤ f
n
(x) ≤ M, for all x in
[a,b], then Converges Uniformly in [a,b].
 
convergentAa
xfaxu
n
nnn
,



n
n
xu
    bainconvergentuniformlyisxu
xd
d
baincontinuousarexu
xd
d
andxu
n
nnn ,&,
1


Return to the Table of Content
Uniformly convergent series have three particular useful properties:
1.If the individual terms u
n
(x) are continuous, the series sum
is also continuous.
2. If the individual terms u
n
(x) are continuous, the series may be integrated term by term.
The sum of the integrals is equal to to the integral of the sum, then
3.The derivative of the series sum f (x) equals the sum of the individual term derivatives

providing the following conditions are satisfied
 



1n
n
xuxf
 



1n
b
a
n
b
a
xdxuxdxf
 



1n
n
xu
xd
d
xf
xd
d

83. SOLO
Complex Variables
Uniformly Convergent Series of Analytic Functions
Suppose that
(i)Each number of a sequence of functions u
1
(z), u
2
(z),…,u
n
(z),…
is Analytic inside a Region D,
(ii)The Series
is Uniformly Convergent through Every Region D’ interior to D.
Then the function

is Analytic inside D, and all its Derivatives can be calculated by term-by-term
Differentiation.


1n
nzu
 



1n
n
zuzf
C
x
y
R
z0
z
w
r
P0
D
Proof:
Let C be a simple closed contour entirely inside D, and
let z a Point inside D. Since u
n(z) is Analytic inside D, we
have:





C
n
n wd
zw
wu
i
zu
2
1
for each function u
n
(z). Hence
 

 






11
2
1
n
C
n
n
n
wd
zw
wu
i
zuzf


84. SOLO
Complex Variables
Uniformly Convergent Series of Analytic Functions
C
x
y
R
z0
z
w
r
P0
D
Proof (continue – 1):
Since is Uniformly Convergent on C, we may multiply by 1/(w-z)
and integrate term-by-term:
and we obtain
 

 






11
2
1
n
C
n
n
n wd
zw
wu
i
zuzf



1n
n
zu
 








11 n
C
n
C
n
n
wd
zw
wu
wd
zw
wu

 







CC
n
n
wd
zw
wf
i
wd
zw
wu
i
zf
 2
1
2
1
1
The last integral proves that f(z) is Analytic inside C, and since C is an
arbitrary closed contour inside D, f(z) is Analytic inside D.

85. SOLO
Complex Variables
Uniformly Convergent Series of Analytic Functions
C
x
y
R
z0
z
w
r
P0
D
Proof (continue – 2):
Since f(z) is Analytic in D, the same is true for f’(z),
therefore we can write






C
wd
zw
wf
i
zf
2
2
1
'

Therefore
q.e.d.







 














11 2
1 22
'
2
1
2
1
2
1
'
n
n
n
C
n
eConvergenc
Uniform
C
n
n
C
zuwd
zw
wu
i
zw
wd
wu
i
wd
zw
wf
i
zf


Hence the Series can be Differentiate term-by-term

86. SOLO
Complex Variables
Uniformly Convergent Series of Analytic Functions
C
x
y
R
z0
z
w
r
P0
D
Remarks on the above Theorem
(i)The contrast between the conditions for term-by-term differentiation of Real Series,
and of Series of Analytic Functions is that
- In the case of Real Series we have to assume that the Differentiated Series is
Uniformly Convergent.
- In the case of Series of Analytic Series the Theorem proved that the Differentiated
Series is Uniformly Convergent.
(ii)If we merely assumed that the given Series is Uniformly Convergent on a certain
Closed Curve C, we could prove as before that f(z) is Analytic at all points inside C.
(iii) Even if we assume that each u
n
(z) is Analytic on the Boundary of the Domain D,
and the Series is Uniformly Convergent on the Boundary, we can not prove that f(z) is
Analytic on the Boundary, or the Differentiated Series Converges on the Boundary.
(iv) The Theorem may be stated as a Theorem on Sequences of Functions:
If f
n(z) is Analytic in D for each value of n, and tends to f(z) Uniformly in any
Region interior to D, then f(z) is Analytic inside D, and f
n
’(z) tends to f’(z) Uniformly in
any Region interior to D.
Return to the Table of Content

87. 87
SOLO
Complex Variables
Let f (z) be analytic at all points within a circle C
0
with center at z
0
and radius r
0
.
Then at each point z inside C
0
:
Taylor’s Series
 

 


  
n
n
zz
n
zf
zz
zf
zzzfzfzf
0
02
0
0
000
!!2
''
'
Power Series
Brook Taylor
1685 - 1731
a convergent power series for some |z-z0|<R (radius of convergence).
C
x
y
R
z
0
C
0
C
1
z
z'
r
0
r
1
r
Proof:
Start with the Cauchy’s Integral Formula:




C
zd
zz
zf
i
zf '
'
'
2
1

Use the identity:



 



1
1
1
1
12
n
n

for:
  


















































 nn
zz
zz
zz
zzzz
zz
zz
zz
zz
zz
zzzzzzzzzz
0
0
0
0
1
0
0
0
0
0
0
0000
'
'
1
1
''
1
'
1
'
1
1
'
1
'
1
'
1

Since z inside C
0
|z-z
0
|=r < r
0
. For z’ is on C
1
we have |z’-z
0
|=r
1
<r
0

88. 88
SOLO
Complex Variables
Taylor’s Series (continue - 1)
Power Series
C
x
y
R
z
0
C
0
C
1
z
z'
r
0
r
1
r
Proof (continue - 1):
Using the Cauchy’s Integral Formula:





C
zd
zz
zf
i
zf '
'
'
2
1













































 nn
zz
zz
zz
zzzz
zz
zz
zz
zz
zf
zz
zf
0
0
0
0
1
0
0
0
0
0 '
'
1
1
''
1
'
'
'
'




 

 

 


 
  
 
 


 
n
n
n
R
C
n
n
n
nzf
C
n
zf
C
zf
C
Rzz
n
zf
zzzfzf
zzzz
zdzf
i
zz
zz
zz
zdzf
i
zz
zz
zdzf
izz
zdzf
i
n
n






























0
0
000
0
0
1
0
!/
0
0
!1/'
2
00
!
'
''
''
2
'
''
2
1
'
''
2
1
'
''
2
1
0
0
0
0
0
0
0







We have:


n
i
n
n
C
n
n
n
r
r
rr
Mr
der
rrr
Mr
zzzz
zdzfzz
R
















11
1
2
0
1
110
0
2''
''
2
0




where |f (z)|<M in C
0
and r/r
1
< 1, therefore:
0


n
nR
q.e.d.
0100 ' rrzzrzz 

89. 89
SOLO
Complex Variables
Let f (z) be analytic at all points within a circle C
0
with center at z
0
and radius r
0
.
Then at each point z inside C
0
:
Taylor’s Series (continue – 2)
 

 


  
n
n
zz
n
zf
zz
zf
zzzfzfzf
0
02
0
0
000
!!2
''
'
Power Series
Brook Taylor
1685 - 1731
a convergent power series for some |z-z0|<R (radius of convergence).
C
x
y
R
z
0
C
0
C
1
z
z'
r
0
r
1
r
Proof (continue – 2):



 



0
01
0
1
!
k
k
k
zz
k
zf
zfSuppose the series converges for z=z
1
:




































01
0
1
001
0
0
01
0
0
0
0
1
!!
zz
zz
M
aM
zz
zz
zz
k
zf
zz
k
zf
zf
a
k
k
k
k
k
k
k
k
k
Since the series converges all its terms are bounded


,2,1,0
!
01
0
 nMzz
k
zf
k
k
Define:
01
0
:
zz
zz
a



Therefore the series f (z) converges for all
010 zzzz 
The region of convergence of a Taylor series of f (z) around a point z
0
is a circle centered
at z
0
and radius of convergence R that extends until f (z) stops to be analytic.

90. 90
SOLO
Complex Variables
Taylor’s Series (continue – 3)




 
n
n
z
n
f
z
f
zffzf
!
0
!2
0''
0'0
2
Power Series
Brook Taylor
1685 - 1731
When z
0
= 0 the series is called Maclaurin’s series after
Colin Maclaurin a contemporary of Brook Taylor.
Colin Maclaurin
1698 - 1746
Examples of Taylor’s Series



z
n
z
e
n
n
z
0!

 







z
n
z
z
n
n
n
0
12
1
!12
1sin





z
n
z
z
n
n
n
0
2
!2
1cos
 






z
n
z
z
n
n
0
12
!12
sinh




z
n
z
z
n
n
0
2
!2
cosh
 11
1
1
0





zz
z
n
nn
Return to the Table of Content

91. 91
SOLO
Complex Variables
Laurent’s Series (1843)
Power Series
If f (z) is analytic inside and on the boundary of the ring
shaped region R bounded by two concentric circles C
1
and
C
2 with center at z
0 and respective radii r
1 and r
2 (r
1 > r
2),
then for all z in R:
Pierre Alphonse Laurent
1813 - 1854
C
1
x
y
R
C
2R
2
R
1
z
0
z
z'
r
P
1
P
0
z'
  
 





 

1 00
0
n
n
n
n
n
n
zz
a
zzazf

 
,2,1,0'
'
'
2
1
2
1
0




 nzd
zz
zf
i
a
C
nn


 
,2,1,0'
'
'
2
1
1
1
0





nzd
zz
zf
i
a
C
nn

Proof:
Since z is inside R we have R
1 <|z-z
0|=r < R
2 , and |z’-z0|= R
1 on C
1 and R
2 on C
2.
Start with the Cauchy’s Integral Formula:

   

 













212
0
1
1
01
'
'
'
'
'
'
'
'
'
'
'
'
'
'
'
'
'
'
0
CCC
P
P
P
PC
dz
zz
zf
dz
zz
zf
zfdzdz
zz
zf
dz
zz
zf
dz
zz
zf
dz
zz
zf
zf


92. 92
SOLO
Complex Variables
Laurent’s Series (continue - 1)
Power Series
Pierre Alphonse Laurent
1813 - 1854
C
1
x
y
R
C
2R
2
R
1
z
0
z
z'
r
Proof (continue – 1):
Since z and z’ are inside R we have R
1
>|z-z
0
|=r >R
2
, |z’-z
0
|=R
1
.
From Cauchy’s Integral Formula:

 





21
'
'
'
'
'
'
CC
dz
zz
zf
dz
zz
zf
zf
Use the identity:



 



1
1
1
1
12
n
n

For I integral:
















































 nn
zz
zz
zz
zzzz
zz
zz
zz
zz
zz
zzzzzz
0
0
0
0
1
0
0
0
0
0
0
00 '
'
1
1
''
1
'
1
'
1
1
'
1
'
1




 

 

 

 
  
 
   
n
n
n
R
C
n
n
n
zs
C
n
za
C
za
C
Rzzzazzzaza
zzzz
zdzf
i
zz
zz
zz
zdzf
i
zz
zz
zdzf
izz
zdzf
i
n
n






























0000100
0
0
1
0
0
02
00
2
0
2
01
2
00
2
''
''
2
'
''
2
1
'
''
2
1
'
''
2
1










1
'
'
'
2
1
C
zd
zz
zf
i
We have:

 
n
n
n
C
n
n
n
R
r
rR
MR
dR
rRR
Mr
zzzz
zdzfzz
R
















11
1
2
0
1
110
0
2''
''
2
0



where |f (z)|<M in R and r/R
1
< 1, therefore:
0


n
nR

93. 93
SOLO
Complex Variables
Laurent’s Series (continue - 2)
Power Series
Pierre Alphonse Laurent
1813 - 1854
C
1
x
y
R
C
2R
2
R
1
z
0
z
z'
r
Proof (continue – 1):
Since z and z’ are inside R we have R
1
>|z-z
0
|=r > R2, |z’-z0|=R
2
.
From Cauchy’s Integral Formula:

 





21
'
'
'
'
'
'
CC
dz
zz
zf
dz
zz
zf
zf
Use the identity:



 



1
1
1
1
12
n
n

For II integral:

















































 nn
zz
zz
zz
zzzz
zz
zz
zz
zz
zz
zzzzzz
0
0
0
0
1
0
0
0
0
0
0
00
'
'
1
1''
1
1
'
1
11
'
1



 

 

 

 
 
 
   
n
n
n
R
C
n
n
n
za
C
n
za
CC
Rzzzazzza
zzzz
zdzfzz
i
zzzz
zdzf
izzzz
zdzf
i
zdzf
i
n
n


































1
001
1
001
0
0
1
0
1
00
2
0
0
0
01
0
01
00
'
'''
2
1
1
'
''
2
11
'
''
2
1
''
2
1










C
zd
zz
zf
i
'
'
'
2
1

We have:

 
n
n
n
C
n
n
n
r
R
rR
RM
dR
rRr
MR
zzzz
zdzfzz
R

















2
2
2
2
0
2
2
2
0
0
2'
'''
2
1
0



where |f (z)|<M in R and R
2
/r< 1, therefore:
0


n
nR
Return to the Table of Content

94. 94
SOLO
Complex Variables
Zeros of Holomorphic Functions






00
00
1
0
1
0


zfandzfzfzf
kk

We say that Holomorphic Function f (z) has a Zero of Order k at z = z
0
if
If f (z) has a Zero of Order k at z = z
0
, by Taylor expansion, we can write
with Holomorphic and nonzero.
zfzzzf
k
k
0



kk
zz
zf
zf
0
:


Note:
(1) For g (z) = 1/ f (z) the Order k Zeros of f (z) are Order k Poles of g (z)

zfzzzf
zg
k
k
111
0


(2) For




zf
zf
zz
k
zf
zf
zf
zd
d
k
k
''
ln
0



z = z
0
, is a Simple Pole.
Return to the Table of Content

95. 95
SOLO
Complex Variables
Theorem: f(z) Analytic and Nonzero → ln|f(z)| Harmonic
If f (z) is analytic for in an Open Set Ω, and has no zeros in Ω,
then ln |f(z)| is Harmonic in Ω.
Proof :
Since f (z) is analytic and has no zeros the logarithm of f(z) is also Analytic
g (z) := ln f (z) is Analytic
Therefore
q.e.d.

    zgizgzgizgzg
eeeezf
ImReImRe


and
 Harmoniczgizgzg )(Im)(Re 

  zgzgizg
eeezf
Re
1
ImRe




Harmoniczgezf
zg
Relnln
Re

meaning
Harmonicyixgyixg
yixg
y
yixg
x
yixg
y
yixg
x
)(Re),(Re
0)(Im)(Im&0)(Re)(Re
2
2
2
2
2
2
2
2













Return to the Table of Content

96. 96
SOLO
Complex Variables
Polynomial Theorem
If f (z) is analytic for all finite values of z, and as |z| → ∞, and
then f (z) is a polynomial of degree ≤ k.
Proof :
Integrating this result we obtain
q.e.d.
 kgivenAsomeforzforzAzf
k
&0, 
Using Taylor Series around any analytic point z = a





 

 af
n
az
afazafzf
n
n
!
1






 Aafaz
z
az
n
af
z
az
k
af
z
az
z
af
z
zf
n
kn
k
k
k
k
k
kkk









!
1
!
1
1
If |z| → ∞ , the previous equation is possible only if f
(n)
(a)=0 for all n > k and all a.
Therefore f
(k)
(a) = constant for all a, i.e. f
(k)
(z)=a
k=constant.

0
1
1
azazazf
k
k
k
k




Continuing to Integrate we obtain


1
1



kk
k
azazf
Return to the Table of Content

97. 97
SOLO
Complex Variables
The Argument Theorem
If f (z) is analytic inside and on a simple closed curve C except for a
finite number of poles inside C
(this is called a Meromorphic Function), then


PNdz
zf
zf
i
C


'
2
1

where N and P are respectively the number of zeros and poles inside C.
Proof:
Let write f (z) as:
 
 
zG
z
z
zf
j
p
j
k
n
k
j
k







where: , 

j
j
k
k pPnN &
     zGzpznzf
jjk
k
k lnlnlnln  

Differentiate this equation:

    



Cinanalytic
j
j
k k
k
zG
zG
z
p
z
n
zf
zf
'
'




 


    


PNpn
zG
zG
iz
p
iz
n
i
dz
zf
zf
i
j
k
k
C
p
C j
j
k
n
C k
k
C
jk









0
'
2
1
2
1
2
1'
2
1

and G (z) ≠ 0 and analytic in C (G’ (z) exists).
   
k
C k
k
C k
k
ndz
z
n
i
dz
z
n
i
k



 

 2
1
2
1
   
j
C j
j
C j
j
pdz
z
p
i
dz
z
p
i
j



 

 2
1
2
1
x
yC
R
1
3

2
k
1
2
j

k
C

3
C
2C
2
C
1
C
j
C

q.e.d.
Return to the Table of Content

98. 98
SOLO
Complex Variables
Rouché’s Theorem
Eugène Rouché
1832 - 1910
If f (z) and g (z) are analytic inside and on a simple closed curve C
and if |g (z)| < |f (z)| on C, then f (z) + g (z) and f (z) have the same number
of zeros inside C.
Proof:
Let F (z):= g (z)/f (z)
If N
1
and N
2
are the number of zeros inside C of f (z) + g (z) and f (z) respectively, and
using the fact that those functions are analytic and C, therefore they have no poles
inside C, using the Argument Theorem we have



C
dz
zf
zf
i
N
'
2
1
2






C
dz
zgzf
zgzf
i
N
''
2
1
1



 
0
1'
2
1
1
'
2
1'
2
1
1
''
2
1
'
2
1
1
'1'
2
1'
2
1'''
2
1
32
21




















CCCC
CCCC
dzFFFF
i
dz
F
F
i
dz
f
f
i
dz
F
F
f
f
i
dz
f
f
i
dz
Ff
FfFf
i
dz
f
f
i
dz
Fff
FfFff
i
NN



We used the fact that |F|=|g/f|<1 on C, so the series 1-F+F
2
+
…
is uniformly
convergent on C and integration term by term yields the value zero. Thus N
1
=N
2
q.e.d.
Return to the Table of Content

99. 99
SOLO
Complex Variables
Foundamental Theorem of Algebra (using Rouché’s Theorem)
Every polynomial equation P (z) = a
0
+ a
1
z+a
2
z
2
+…+a
n
z
n
=0 with degree
n ≥ 1 and a
n
≠ 0 (a
i
are complex constants) has at exactely n zeros.
Proof:
Define:
Take C as the circle with the center at the origin and radius r > 1.
q.e.d.

n
n
zazf: 
1
1
2
210
:



n
n
zazazaazg 


ra
aaaa
ra
rararara
ra
rararaa
za
zazazaa
zf
zg
n
n
n
n
n
n
nnn
n
n
n
n
n
n
n
n
1210
1
1
1
2
1
1
1
0
1
1
2
210
1
1
2
210


















By choosing r large enough we can make |g (z)|/|f (z)|<1, and using Rouché’s Theorem

n
n
n
n
zazazazaazgzfzP 


1
1
2
210

and (n zeros at the origin z = 0) have the same number of zeros,
i.e. P (z) has exactly n zeros.

n
n
zazf:
Return to the Table of Content

100. 100
SOLO
Complex Variables
Jensen’s Formula
Johan Ludwig William Valdemar
Jensen
(1859 – 5 1925)
C
x
y
r
1
a
n
a
D

which is the Mean-Value Property of the Harmonic Function ln |f(z)|.
Suppose that ƒ is an Analytic Function in a region in the
complex plane which contains the closed disk D of radius r
about the origin, a
1
,
 a
2
,
 ..., a
n
are the zeros of ƒ in the interior
of D repeated according to multiplicity, and ƒ(0)
 ≠ 0. Jensen's
formula states that
This formula establishes a connection between the Moduli of the
zeros of the function ƒ(z) inside the disk D and the average of
log |f(z)| on the boundary circle |z|
 = r, and can be seen as a
generalization of the Mean Value Property of Harmonic
Functions. Namely, if f(z) has no zeros in D, then Jensen's
formula reduces to
  














2
01
ln
2
1
ln0ln derf
r
a
f
j
n
k
k
 





2
0
ln
2
1
0ln derff
j

101. 101
SOLO
Complex Variables
Jensen’s Formula (continue – 1)
Proof
If f has no Zeros in D, then we can use Gauss’ Main Value
Theorem to ln f(z) that is Harmonic in D
 





2
0
ln
2
1
0ln derff
j
Since f has Zeros a
1
, a
2
,…, a
n
inside D, ( |z| < r) let define the Holomorphic Function
F (z) without Zeros in D :

 




n
k k
k
raz
rza
zfzF
1
2
/
/1
:
Apply the Gauss’ Main Value theorem for |z|=r
    


















2
0
2
01
ln
2
1
ln
2
1
ln0ln0ln derfderF
r
a
fF
ii
n
k
k
and:


1
1
1
1
/1
1
/1
/
/1
/
/1
2
2
1
2
22
1
22
2
2
















r
z
a
r
z
a
r
zz
z
a
rza
z
a
rza
z
r
raz
rza
raz
rza
rzzz
k
k
k
k
k
k
k
k
k
k
The Zeros of f(z) are cancelled.
q.e.d.
We have: i.e. z
k1
is outside the Disk
D.
rzarzforrza
k
ra
kkkk
k


1
2
1
2
1
/0/1
C
x
y
D
r
z'
r

k
a
ka
r

kk
arz /
2


102. 102
SOLO
Complex Variables
Jensen’s Formula (continue – 2)
Johan Ludwig William Valdemar
Jensen
(1859 – 5 1925)


zh
zg
zzf
l

Jensen's formula may be generalized for functions which are
merely meromorphic on D. Namely, assume that
where g and h are analytic functions in D having zeros at
respectively, then Jensen's formula for meromorphic functions
states that
Jensen's formula can be used to estimate the number of zeros of
analytic function in a circle. Namely, if f is a function analytic in a
disk of radius R centered at z
0 and if |f(z)| is bounded by M on the
boundary of that disk, then the number of zeros of f(z) in a circle of
radius r < R centered at the same point z
0
does not exceed
 0\,,0\,,
11
DbbandDaa
mn
 








2
01
1
ln
2
1
ln
0
0
ln
i
m
nnm
erf
bb
aa
r
h
g



0
ln
/ln
1
zf
M
rR
C
x
y
r
1
a
n
a
D


103. 103
SOLO
Complex Variables
Jensen’s Formula (continue – 3)
Johan Ludwig William Valdemar
Jensen
(1859 – 5 1925)
Jensen's formula may be put in an other way. If n (r) denotes
the number of Zeros, including multiplicity, and p (r) denotes
the number of Poles, including multiplicity, for |z| < r, then
the Jensen’s Formula can be written as
C
x
y
r
1
a
na
D


 0lnln
2
1
lnln
2
0110
fderf
b
r
a
r
dx
x
xpxn
j
n
j j
m
k k
r


























Proof




























n
j
j
m
k
k
n
j j
m
k k
brnarm
b
r
a
r
1111
lnlnlnlnlnln
     








 





 n
n
j
jjm
m
k
kk
brnbbjarnaak lnlnlnlnlnlnlnln
1
1
1
1
1
1

















 




 r
r
n
j
r
r
r
r
m
k
r
r
n
j
jn
k
k
x
xd
n
x
xd
j
x
xd
m
x
xd
k
1
1
1
1
11

104. 104
SOLO
Complex Variables
Jensen’s Formula (continue – 4)
Johan Ludwig William
Valdemar Jensen
(1859 – 5 1925)
Jensen's formula may be put in an other way. If n (r) denotes
the number of Zeros, including multiplicity, and p (r) denotes
the number of Poles, including multiplicity, for |z| < r, then
the Jensen’s Formula can be written as
C
x
y
r
1
a
n
a
D


 0lnln
2
1
lnln
2
0110
fderf
b
r
a
r
dx
x
xpxn
j
n
j j
m
k k
r


























Proof (continue – 1)








































 r
r
n
j
r
r
r
r
m
k
r
r
n
j j
m
k k
n
j
jn
k
k
x
xd
n
x
xd
j
x
xd
m
x
xd
k
b
r
a
r
1
1
1
111
11
lnln
But k = n (x) for r
m
≤ x ≤ r
m+1
, m = n (x) for r
n
≤ x ≤ r, and
j = p (x) for r
j
≤ x ≤ r
j+1
, n = p (x) for r
n
≤ x ≤ r Hence
 
xd
x
xp
xd
x
xn
b
r
a
r
rrn
j j
m
k k
 

















 0011
lnln
q.e.d.
Return to the Table of Content

105. 105
SOLO
Complex Variables
Poisson-Jensen’s Formula for a Disk
Johan Ludwig William Valdemar
Jensen
(1859 – 5 1925)
Poisson Formula states:
Siméon Denis Poisson
1781-1840
C
x
y
R
1
a
na
D

Let g (z) be analytic inside and on the circle C defined by
|z| = R, and let z = r e
iθ
be any point inside C, then:
 









2
0
2
22
2
1
deRg
ereR
rR
erg
i
ii
i
In our case ƒ is an analytic function in a region in the complex
plane which contains the closed disk D of radius R about the
origin, a
1,
 a
2,
 ..., a
m are the Zeros, and b
1,
 b
2,
 ..., b
n are the Poles
of ƒ in the interior of D repeated according to multiplicity.
Since f has Zeros a
1
, a
2
,…, a
n
inside D, ( |z| < r)
let define the Holomorphic Function
F (z) without Zeros in D :

 

Rz
Rzb
Rbz
Raz
Rza
zfzF
m
k
n
j j
j
k
k






 
 1 1
2
2
/1
/
/
/1
:
Zeros and Poles of f(z) are cancelled, and new ones are outside D.

106. 106
SOLO
Complex Variables
Poisson-Jensen’s Formula for a Disk
Johan Ludwig William Valdemar
Jensen
(1859 – 5 1925)
Let apply the Poisson Formula to g (z) = ln |F (z)|:
  
 


 





n
j j
j
m
k k
k
Rzb
Rbz
Raz
Rza
zfzFzg
1
2
1
2
/1
/
ln
/
/1
lnlnln:
 

1
/1
/
/
/1
2
2 Rz
j
j
Rz
k
k
Rzb
Rbz
Raz
Rza







Siméon Denis Poisson
1781-1840
C
x
y
R
1
a
na
D

We proved that:
 
 ii
eRfeRg ln
The Poisson Formula to g (z) = ln F (z) is:

 

 RzdeRf
ereR
rR
Rzb
Rbz
Raz
Rza
zf
i
ii
n
j j
j
m
k k
k


















2
0
2
22
1
2
1
2
ln
2
1
/1
/
ln
/
/1
lnln
RzRzbandRzRza
jjjkkk

1
2
11
2
1
0/1,0/1

107. 107
SOLO
Complex Variables
Poisson-Jensen’s Formula for a Disk
Johan Ludwig William Valdemar
Jensen
(1859 – 5 1925)
Siméon Denis Poisson
1781-1840
C
x
y
R
1
a
na
D

The Poisson-Jensen’s Formula for a Disk is:

 

 RzdeRf
ereR
rR
Rzb
Rbz
Raz
Rza
zf
i
ii
n
j j
j
m
k k
k


















2
0
2
22
1
2
1
2
ln
2
1
/1
/
ln
/
/1
lnln
For z = r = 0 we obtain the Jensen’s Formula:
 













2
021
21
ln
2
1
ln0ln deRfR
aaa
bbb
f
inm
m
n


If there are no Zeros or Poles in D, it reduces to
Poisson’s Formula:
  RzdeRf
ereR
rR
zf
i
ii









2
0
2
22
ln
2
1
ln
Return to the Table of Content

108. 108
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
If f (z) is analytic inside and on the boundary of a circle C
,
except it’s center z
0,
then
according to Laurent’s Series:
C
x
y
R
R
z
0
z
z'
r
P
0
  
 





 

1 00
0
n
n
n
n
n
n
zz
a
zzazf

 
,2,1,0'
'
'
2
1
1
0





nzd
zz
zf
i
a
C
nn


 
,2,1,0'
'
'
2
1
1
0





nzd
zz
zf
i
a
C
nn

Let compute
  

 
,2,1,0'
'
'
''''
1 00
0 

  





nzd
zz
zf
azdzzazdzf
n
C
nn
C
n
C
n
n
 

  








12
10
'
'
'
&,2,1,00''
0
0
ni
n
zd
zz
zf
nzdzz
C
n
C
n


Therefore: 
1
2''

aizdzf
C

Because only a
-1 is involved in the integral above, it is called the residue of
f (z) at z = z
0
.
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109. 109
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
According to residue definition the residue of f (z) at z = z
0 can be computed as follows:
C
x
y
R
R
z
0
z
z'
r
P
0
If z = z
0
is a pole of order k, i.e. the Laurent series at
z
0
is
Then:




C
zdzf
i
a ''
2
1
1

Calculation of the Residues
 
 







2
02010
0
2
0
2
0
1
zzazzaa
zz
a
zz
a
zz
a
k
k

 zfzz
zd
d
k
a
k
k
k
zz
01
1
1
!1
1
lim
0







    






2
02
1
0100
2
02
1
010
kkk
k
kkk
zzazzazzaazzazzazfzz
and:
If z = z
0
is a pole of order k=1, then:
 zfzza
zz
01
0
lim


 





 

k
n
n
n
n
n
n
zz
a
zzazf
1 00
0
Return to the Table of Content

110. 110
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
The Residue Theorem
If f (z) is analytic inside and on the boundary of a closed curve C
,
except at the singularities
z
01
, z
02
,…,z
0n
, which have residues Re
1
, Re
2
,…,Re
n
, then:
Proof:
  
n
C
izdzf ReReRe2''
21 


x
yC
R
01z
n
z
C
2zC
0 2z
1
z
C
nz
0Surround every singularity z
0i by a small closed curve
C
zi
, that enclosed only this singularity. Connect those
Curves to C by a small corridor (the width of which
shrinks to zero, so that the integration along the opposite
directions will cancel out)
    0''''''''
21


znzz
CCCC
zdzfzdzfzdzfzdzf 
We have , therefore:
i
C
izdzf
zi
Re2'' 


     
n
CCCC
izdzfzdzfzdzfzdzf
znzz
ReReRe2''''''''
21
21


 
q.e.d.
Return to the Table of Content

111. 111
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Evaluation of Integrals
Theorem 1
If |F (z)| ≤ M/R
k
for z = R e
iθ
where k > 1 and M are constants, then
where Γ is the semicircle arc of radius R, center at origin, in the
upper part of z plane.
 0lim 



zdzF
R
x
y

R
Proof:

1
0
1
0





kk
i
k
eRz
k
R
M
d
R
M
deRi
R
M
zd
R
M
zdzF
i





Therefore:  0limlim
1
1
0
1








k
k
R
k
R
R
M
d
R
M
zdzF



 0lim 



zdzF
R
and:    0limlimlim0 







zdzFzdzFzdzF
RRR
q.e.d.
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112. 112
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Evaluation of Integrals
Jordan’s Lemma
If |F (z)| ≤ M/R
k
for z = R e
iθ
where k > 0 and M are constants, then
where Γ is the semicircle arc of radius R, center at origin, in the
upper part of z plane, and m is a positive constant.
 0lim 



zdzFe
zmi
R
x
y

R
Proof:
 0lim 



zdzFe
zmi
R
using:
q.e.d.
 









0
deRieRFezdzFe
iieRmi
eRz
zmi
i
i
  











2/
0
sin
1
0
sin
1
0
sin
0
sincos
00
2















dRe
R
M
dRe
R
M
dReRFe
deRieRFedeRieRFedeRieRFe
Rm
k
Rm
k
iRm
iiRmRmiiieRmiiieRmi
ii
2/0/2sin   for

2/
1
sin
/2 
  
Rm
k
Rm
k
Rm
k
iieRmi
e
R
M
de
R
M
de
R
M
deRieRFe
i






1
222
2/
0
/2
1
2/
0
sin
1
0








  01
2
limlim
0



Rm
k
R
iieRmi
R
e
R
M
deRieRFe
i




Marie Ennemond Camille Jordan
1838 - 1922

113. 113
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Evaluation of Integrals
Jordan’s Lemma Generalization
If |F (z)| ≤ M/R
k
for z = R e
iθ
where k > 0 and M are constants, then
for Γ a semicircle arc of radius R, and center at origin:
 00lim 



mzdzFe
zmi
R
x
y

R
where Γ is the semicircle, in the upper part of z plane.
1
 00lim 



mzdzFe
zmi
R
x
y

R
where Γ is the semicircle, in the down part of z plane.
2
 00lim 



mzdzFe
zm
R x
y

R
where Γ is the semicircle, in the right part of z plane.
3
 00lim 



mzdzFe
zm
R
where Γ is the semicircle, in the left part of z plane.
4
x
y

R
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114. 114
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Evaluation of Integrals
Integral of the Type Bromwwich-Wagner 



jc
jc
ts
sdsFe
i2
1
The contour from c - i ∞ to c + i ∞ is called Bromwich Contour
Thomas Bromwich
1875 - 1929
x
y
0
t
R
c
x
y
0
t
R
c
   




























0
0
2
1
lim
2
1
2
1
tzFeRes
tzFeRes
zdzF
i
sdsFesdsFe
i
sdsFe
i
tf
tz
planezRight
tz
planezLeft
ts
ic
ic
ts
R
ic
ic
ts


where Γ is the semicircle, in the right part of z plane, for t < 0.
where Γ is the semicircle, in the left part of z plane, for t > 0.
This integral is also the Inverse Laplace Transform.
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115. 115
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Evaluation of Integrals
Integral of the Type ,F (sin θ, cos θ) is a rational function of
sin θ and cos θ
 



2
0
cos,sin dF
Let z = e
iθ
22
cos,
22
sin
11 








zzee
i
zz
i
ee
iiii 

zizdddzideizd
i
/ 

 
 






 


C
zi
zdzz
i
zz
FdF
2
,
2
cos,sin
11
2
0


where C is the unit circle with center at the origin.
C
x
y
R=1
Return to the Table of Content

116. 116
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Evaluation of Integrals
Definite Integrals of the Type .




xdxF
If the conditions of Theorem 1, i.e.:
if |F (z)| ≤ M/R
k
for z = R e
iθ
where k > 1 and M are constants, then
and we can write
 0lim 



zdzF
R
x
y

R
    zFResizdzFzdzFxdxFxdxF
planezUpper
R
R
R
2lim 
















Example: Heaviside Step Function
x
e
i
xF
txi
2
1
:
x
y

R
0t
x
y

R
0t
This function has a single pole at z = 0.
For t > 0 Γ is the semicircle, in the upper part of z plane.
We also include on the path x = - ∞ to x = + ∞ a small
semicircle such that the pole z = 0 is included.
For t < 0 Γ is the semicircle, in the lower part of z plane.
We also include on the path x = - ∞ to x = + ∞ a small
semicircle such that the pole z = 0 is excluded.











00/
01/
2
1
tzeRes
tzeRes
xd
x
e
i
tzi
planezLower
tzi
planezUpper
txi

Return to the Table of Content

117. 117
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Evaluation of Integrals
Cauchy’s Principal Value
Cauchy’s Principal value deals with integrals that have singularities along the
integration paths. Start with the following:
Theorem 1:
If f (z) is analytic on and inside a positive-sensed circle C of radius ε, centered at
z = z
0
, then


0
0
0
lim zfi
zz
zdzf
C





where C
ψ
is every arc on C of angle ψ.
Proof:
Since f (z) is analytic inside and on C we can use
the Taylor series expansion to write







1
0
0
0
!
n
n
n
zz
n
zf
zfzf









zg
n
n
n
zz
n
zf
zz
zf
zz
zf








1
1
0
0
0
0
0 !
Consider the integral on C
ψ defined by z=z
0 + ε e
iθ
θ
0 ≤ θ ≤ θ
0 + ψ
C
x
y

C
0
z
0
0


O

118. 118
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Evaluation of Integrals
Cauchy’s Principal Value (continue – 1)
Proof (continue – 1):
Since g (z) is bounded inside and on C , there is a positive number M such that
|g (z)| < M for all z such that |z – z
0
| < ε
 






 CCC
zdzg
zz
zdzf
zz
zdzf
0
0
0

  
00
0
0
0
0
0
0
0
zfidizf
zz
zd
zf
zz
zdzf
i
ezz
CC









 


  

MLMzdzgzdzg
CC


Where L = ψ ε is the length of C
ψ.
   0lim0limlim
000

 



CC
zdzgMzdzg
    0lim0limlimlim0
0000

 


CCCC
zdzgzdzgzdzgzdzg
Therefore


0
0
0
lim zfi
zz
zdzf
C





q.e.d.
Note: For ψ = 2 π we recover the Cauchy’s Integral result.
C
x
y
C
0
z
0
0


O

119. 119
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Evaluation of Integrals
Cauchy’s Principal Value (continue – 2)
Theorem 2:
If F (z) is analytic on and inside a positive-sensed circle C of radius ε, except at the
center of C, z = z
0, that is a simple pole of F (z), then
C
x
y

C
0
z
0
0


O
  
0
0
lim zFResizdzF
C





where C
ψ
is every arc on C of angle ψ.
Proof:
Since f (z) is analytic inside and on C by using Theorem 1 we obtain the
desired result.
The function:


   









000
00
0
lim zzzFzzzFRes
zzzFzz
zf
zz

120. 120
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Evaluation of Integrals
Cauchy’s Principal Value (continue – 3)
Theorem 3:
If F (z) is analytic on and inside a positive-sensed curve C, except at the interior poles
z
int1, z
int2,…, z
int n and the simple poles on the curve C, z
cont1, z
cont2,…, z
cont m,
then
Proof:
x
y
1in t
z
2i nt
z
nz
int
1sc ont
ext
z
2c
z
c kz
1c

2c

0C
2sc o nt
ext
z
1
in tc on tz
2
in tc o nt
z
mc on t
ext
z
sc ontz
in t
sc on t
in t

mc ont
ext

2
in tc ont
2sc ont
ext

1sc o nt
ext

2
in tc ont
Cz
2sc o nt
ext
Cz
1sc on t
ext
Cz
1
in tc o nt
Cz
mc o nt
ext
Cz
sc o nt
Cz
in t



m
k
kcontCCC
1
0
       










 

s
k
kcont
n
j
j
m
sk
C
s
k
CC
zFReszFResizdzFzdzFzdzF
kex tcontkcont
1
int
1
int
11
2
int0

    










 

m
k
kcont
n
j
j
C
zFReszFResizdzF
11
int
2
1
2
Let encircle the simple poles z
cont k
on the C contour by
semicircles C
cont k
of radiuses ε cont k such that,
randomly, z
cont int 1,…, z
cont int s, are inside the
integration contour, and z
cont ext s+1
,…,z
cont ext m
are
outside the integration contour. We have:
where

121. 121
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Evaluation of Integrals
Cauchy’s Principal Value (continue – 4)
Proof (continue – 1):
x
y
1in t
z
2i nt
z
nz
int
1sc ont
ext
z
2c
z
c kz
1c

2c

0C
2sc ont
ext
z
1
in tc on tz
2
in t
c ontz
mc on t
ext
z
sc ontz
in t
sc on t
in t

mc ont
ext

2
in tc o nt

2sc o nt
ext

1sc ont
ext

2
in t
c ontCz
2sc ont
ext
Cz
1sc on t
ext
Cz
1
in tc o ntCz
mc ont
ext
Cz
sc ont
Cz
in t
   









s
k
kcont
s
k
C
zFResizdzF
kcont
kcont
1
int
1
0
int
int
lim 

The integrals along the semicircles C
cont k
of the
singularities z
cont ext s+1
,…,z
cont ext m
that are
outside the integration contour, are in the negative
direction, and we have, according to Theorem 2:
     










 


m
k
kcont
n
j
j
CC
zFReszFResizdzFzdzF
c
11
int
0 2
1
2lim
0


therefore:
Since the integrals along the semicircles C
cont k
of the
singularities z
cont int 1
,…, z
cont int s
, that are inside the
integration contour, are in the positive direction we have,
according to Theorem 2:
   









m
sk
kextcont
m
sk
C
zFResizdzF
kextcont
kcont
11
0
int
lim 

       










 

s
k
kcont
n
j
j
m
sk
C
s
k
CC
zFReszFResizdzFzdzFzdzF
kextcontkcont
1
int
1
int
11
2
int0

Note: This result is independent on the way that we encircled the simple poles on
the curve C.
q.e.d.

122. 122
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Evaluation of Integrals
Cauchy’s Principal Value (continue – 5)
If F (x) is continuous in a ≤ x ≤ b except at a point x0 such that a < x0 < b, then if
ε
1 and ε
2 are positive then the integral exists if and only if the limit:
  
















b
x
x
a
b
a
xdxFxdxFxdxF
10
10
2
1
0
0
lim






b
a
xdxF
 
















b
x
x
a
xdxFxdxF
10
10
2
1
0
0
lim




exists. If the limit does exist, the integral is equal to the value of this limit:
For this limit to exist, it must always have the same definite value regardless of how
the quantities ε1 and ε2 approach zero.
Cauchy’s Principal Value is defined as:   















b
x
x
a
b
a
xdxFxdxFxdxFPV



0
0
0
lim:
Clearly, if the integral exists, then PV exists and is equal to the integral,
but the opposite is not true.


b
a
xdxF

123. 123
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Evaluation of Integrals
Cauchy’s Principal Value (continue – 6)
Example:
x
y
x
1
a
a


a
a
xd
x
1

















a
a
xd
x
xd
x
1
1
2
1
11
lim
0
0




doe’s not exists
0
11
lim
11
lim
11
lim
1
00
0






















































aa
xu
a
a
vu
a
a
a
a
xd
x
xd
x
ud
u
xd
x
vd
v
xd
x
xd
x
PV
Cauchy’s PV does exist, in this case, but has no meaning.
Return to the Table of Content

124. 124
SOLO
Example



0
sin
dk
k
kr
Let compute:
x
y
R

A
B
C
D
E
F
G
H
RxRx
For this use the integral: 0

ABCDEFGHA
zi
dz
z
e
Since z = 0 is outside the region of integration
0


 BCDEF
zi
R
xi
GHA
zi
R
xi
ABCDEFGHA
zi
dz
z
e
dx
x
e
dz
z
e
dx
x
e
dz
z
e


















00
0000
sin
2
sin
2
sin
lim2limlimlim dk
k
rk
idx
x
x
idx
x
x
idx
x
ee
dx
x
e
dx
x
e
R
R
R
xixi
R
R
xi
R
R
xi
R























idideidei
e
e
dz
z
e i
i
i
eii
i
ei
ez
GHA
zi

 


00
1
0
0
00
limlimlim

 01
2
2
0
/2
/2sin
0
sin
00







R
RRReRii
i
eRi
eRz
BCDEF
zi
e
R
dedededeRi
eR
e
dz
z
e i
i
i













Therefore: 0
sin
2
0



idk
k
rk
idz
z
e
ABCDEFGHA
zi 
2
sin
0



dk
k
kr
Complex Variables

125. 125
SOLO
Example 2
0,
1
21
2
1



RRxd
x
R
R
kLet compute: for k integer and positive
x
y
R

A
B
C
D
E
F
G
H
RxRx
This integral has a singularity on the path of
integration on x = 0:
Complex Variables
 
  































1
100
lim
lim
11
lim
1
1
2
1
2
1
1
1
1
0
0
11
0
0
0
0
2
1
2
2
1
1
2
1
2
2
1
1
2
1
2
1
kdefinednot
k
kRkRkk
xkxkxd
x
xd
x
xd
x
kkkk
R
k
R
k
R
k
R
k
R
R
k











Let compute:

 

 















































oddk
k
e
k
kRkRkk
xk
eR
deRi
xkxd
x
zd
z
xd
x
xd
x
ki
k
kkkk
R
k
kik
i
R
k
R
k
C
k
R
k
R
R
k
&10
100
1
1
lim
lim
111
lim
1
2
1
11
2
1
1
1
0
1
0
1
00
2
2
1
1
2
1
2
1















126. 126
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Evaluation of Integrals
Differentiation Under Integral Sign, Leibnitz’s Rule


constantbaxd
xF
xdxF
d
d
b
a
b
a





,
,
,




This is true if a and b is constant, α is real and α
1
≤ α ≤ α
2
where
α
1
and α
2
are constants, and F (x,α) is continuous and has
continuous partial derivative with respect to α for a ≤ x ≤ b,
α
1
≤ α ≤ α
2
.
Gottfried Wilhelm
von Leibniz
(1667-1748)











b
a
d
bd
d
ad
b
a
xd
xF
xdxFxdxF
d
d










,
,,
When a and/or b are functions of α, then:
Return to the Table of Content

127. 127
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Summation of Series
Proof:
Start with the following contour of integration C
N
,
that is a square with vertices at (N+1/2) (±1±i):
x
y
iN 





 1
2
1
12 N 1N
1N N 12
iN 





1
2
1
iN 





1
2
1
iN 





 1
2
1
N
C
We want to show that on C
N:








e
e
z
1
1
cot
For y > 1/2
12
2
:
1
1
1
1
cot
A
e
e
e
e
ee
ee
ee
ee
ee
ee
ee
ee
z
y
y
yy
yy
yxiyxi
yxiyxi
yxiyxi
yxiyxi
zizi
zizi












































For y < - 1/2
12
2
1
1
1
1
cot A
e
e
e
e
ee
ee
ee
ee
z
y
y
yy
yy
yxiyxi
yxiyxi
































 zfzsnf
zfpoles
cotRe




128. 128
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Summation of Series (continue – 1)
Proof (continue – 1):
Start with the following contour of integration C
N,
that is a square with vertices at (N+1/2) (±1±i):
x
y
iN 





 1
2
1
12 N 1N
1N N 12
iN 





1
2
1
iN 





1
2
1
iN 





 1
2
1
N
C
We want to show that on C
N:








e
e
z
1
1
cot
For y > ½ and y < - 1/2
1:
1
1
cot A
e
e
z 








For - ½ ≤ y ≤ ½ consider, first, z = N +1/2 + i y
   











e
e
AAy
yiyiNz
1
1
2/tanhtanh
2/1cot2/1cotcot
12
For - ½ ≤ y ≤ ½ and z = -N -1/2 + i y
  
12
2/tanhtanh2/1cotcot AAyyiNz  
y






e
e
1
1
2
1
zcot
2
1







2
t anh


129. 129
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Summation of Series (continue -2)
Proof (continue -2) :
x
y
iN 





 1
2
1
12 N 1N
1N N 12
iN 





1
2
1
iN 





1
2
1
iN 





 1
2
1
N
C
We proved that on C
N
: A
e
e
z 








1
1
cot
Residue of π cot (π z) f (z) at the poles of cot (π z), i.e.
z = n, n = 0, ±1, ±2, …
Case 1: f (z) has finite number of poles
    




nfnfn
z
zfz
z
nz
zfznzzfzRes
nz
HopitalL
nz
nznz






 











cos
cos
limcos
sin
lim
cotlimcot
'
  


 zfzszfzResdzzfz
N
N
Cin
zfpoles
n
n
nf
nz
C
 cotRecotcot 






 
 048limlimcotlim
48

















N
N
MA
LM
N
A
dzzfz
k
N
NL
Ck
N
C
N
NC
N
N




 zfzsnf
NCin
zfpoles
cotRe




130. 130
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Summation of Series (continue -3)


 zfzsnf
N
Cin
zfpoles
cotRe



Proof (continue -3) :
x
y
iN 





 1
2
1
12 N 1N
1N N 12
iN 





1
2
1
iN 





1
2
1
iN 





 1
2
1
N
C
We proved that on C
N
: A
e
e
z 








1
1
cot
Case 1: f (z) has finite number of poles
  0cotlimcotlim 
 
NN C
N
C
N
dzzfzdzzfz 
 

 zfzsnfdzzfz
N
N
Cin
zfpoles
n
n
C
 cotRecot 



Therefore 

 zfzsnf
zfpoles
cotRe



q.e.d.
Case 2: f (z) has infinite number of poles
Since CN is expanding to include all s plane, when N → ∞, it will encircle, at the
limit all the poles of f (z).
Return to the Table of Content

131. 131
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Summation of Series
Proof:
Start with the same contour of integration C
N
,
that is a square with vertices at (N+1/2) (±1±i):
x
y
iN 





 1
2
1
12 N 1N
1N N 12
iN 





1
2
1
iN 





1
2
1
iN 





 1
2
1
N
C
On C
N:
2csc Az


 zfzsnf
zfpoles
n
cscRe1 



Residue of π csc (π z) f (z) at the poles of csc (π z), i.e.
z = n, n = 0, ±1, ±2, …
    




nfnf
z
zf
z
nz
zfznzzfzRes
n
nz
HopitalL
nz
nznz
1
cos
lim
sin
lim
csclimcsc
'






 









  


 zfzszfzResdzzfz
N
nN
Cin
zfpoles
n
n
nf
nz
C
N
 cscRecsccsclim0
1









 zfzsnf
zfpoles
n
cscRe1 



q.e.d.
Return to the Table of Content

132. 132
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Summation of Series
Proof:
Start with the same contour of integration C
N
,
that is a square with vertices at (N+1/2) (±1±i):
x
y
iN 





 1
2
1
12 N 1N
1N N 12
iN 





1
2
1
iN 





1
2
1
iN 





 1
2
1
N
C
On C
N:
3tan Az

 zfzs
n
f
zfpoles
tanRe
2
12









Residue of π tan (π z) f (z) at the poles of tan(π z), i.e.
z = (2n+1)/2, n = 0, ±1, ±2, …

 


 

 
































 













 



2
12
2
12
cos
limsin
cos
2
12
lim
tan
2
12
limtan
2/12
'
2/12
2/122/12
n
f
n
f
z
zfz
z
n
z
zfz
n
zzfzRes
nz
HopitalL
nz
nznz






  

 zfzszfzResdzzfz
N
N
Cin
zfpoles
n
n
n
f
nz
C
N
 tanRetancsclim0
2
12













 zfzs
n
f
zfpoles
tanRe
2
12









q.e.d.
Return to the Table of Content

133. 133
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Summation of Series
Proof:
Start with the same contour of integration C
N
,
that is a square with vertices at (N+1/2) (±1±i):
x
y
iN 





 1
2
1
12 N 1N
1N N 12
iN 





1
2
1
iN 





1
2
1
iN 





 1
2
1
N
C
On C
N:
4sec Az


 zfzs
n
f
zfpoles
n
secRe
2
12
1 









Residue of π sec (π z) f (z) at the poles of sec (π z), i.e.
z = (2n+1)/2, n = 0, ±1, ±2, …

 


 

 
 































 













 



2
12
1
2
12
sin
lim
cos
2
12
lim
2
12
limsec
2/12
'
2/12
2/122/12
n
f
n
f
z
zf
z
n
z
zfzesc
n
zzfzRes
n
nz
HopitalL
nz
nznz





  


 zfzszfzResdzzfz
N
n
N
Cin
zfpoles
n
n
n
f
nz
C
N
 secResecseclim0
2
12
1















 zfzs
n
f
zfpoles
n
secRe
2
12
1 









q.e.d.
Return to the Table of Content

134. 134
SOLO
Perron’s Formula
 






11
10
2
1
2:Re
aif
aif
ds
s
a
i
ss
s

Oskar Perron
( 1880 – 1975)
1 32

t
10
ads
s
a
LC
s
10
ads
s
a
RC
s
L
C
R
C



R
eRsC
j
L
0cos2
2:





R
eRsC
j
R
0cos
2:


R
Proof
Define the two semi-circular paths
C
L
(left side), C
R
(right side) with s=2 as
the common origin., and R → ∞.






RLRL
RLRL
C
R
C
R
C
i
iRR
C
s
dadR
R
a
deRi
iRR
a
ds
s
a
,,
,,
cos
cos
sincos
sincos




















LR
LR
RLRL
CC
CC
C
R
R
C
s
R aora
aora
dads
s
a
)0cos&1()0cos&1(
)0cos&1()0cos&1(0
limlim
,,
cos




Complex Variables

135. 135
SOLO
Perron’s Formula
 






11
10
2
1
2:Re
aif
aif
ds
s
a
i
ss
s

1 32

t
10
ads
s
a
L
C
s
10
ads
s
a
RC
s
L
C
R
C



R
eRsC
j
L
0cos2
2:





R
eRsC
j
R
0cos
2:


R
Proof (continue)
We can see tat























10Residue
11lim
1Residue
1Residue
2Re
0
2Re
2Re
a
s
a
a
s
a
s
a
s
a
a
s
a
s
Cs
s
s
s
Cs
s
Cs
RR
L
q.e.d.










































1Residue
1Residue
1
1
1
1
2
1
2Re
2Re
2Re
2Re
0
2
22:Re
a
s
a
a
s
a
ads
s
a
ads
s
a
ads
s
a
ads
s
a
ds
s
a
ds
s
a
i
s
Cs
s
Cs
Cs
s
Cs
s
C
s
C
s
i
i
s
ss
s
R
L
R
L
R
L


Complex Variables
Return to the Table of Content

136. 136
SOLO
z
z
ofzerosn
n
z
z
z
n 


 sin
,2,11
sin
1
2
2













z
ofzerosn
e
n
z
ez
zte
n
n
z
z
zt











 



 1
,2,1
1
1
1
0
1


Euler’s Product
  





  












2/1
1
2/2/
1
1
2
0
10
1
2
11
1
1
zzz
n
n
z
zeroszTrivial
zerosztrivialNon
z
zofpole
zba
primep
z
e
n
z
e
z
z
e
pz












 



















Weierstrass Product
Hadamard Product















1
2
2
sin1
1
n
z
z
n
z
z
zz



 
 
















0Im
0
2/12ln
1
2/112






s
e
s
ss
e
s
Infinite Products
Complex Variables

137. 137
SOLO
In 1735 Euler solved the problem, named “Basel Problem” , posed by Mengoli in
1650, by showing that
6
1
4
1
3
1
2
1
1
2
1
2232

 

nn








































1
22
2
2
2
2
2
2
2
1
9
1
4
11
sin
k k
xxxx
x
x


He did this by developing an Infinite Product for sin x /x:
The roots of sin x are x =0, ±π, ±2π, ±3π,…. However sin x/x is not a
polynomial, but Euler assumed (and check it by numerical computation)
that it can be factorized using its roots as
We now that if p (x) and q (x) are two polynomials, then using the roots of the
two polynomials we have:


   
   
m
n
qqqq
pppp
xxxxxxa
xxxxxxa
xq
xp





21
21
We want to show how to express a general solution for complex function f (x)
using the zeros and the poles (finite or infinite) of f (x).
Infinite Products
Complex Variables

138. 138
SOLO
Definition 1:
We say that the Infinite Product converges, if for any N
0
> i
N
the
limit
exists and is nonzero.
If this is satisfied then we can compute
     







N
Nj
i
N
N
Nj
i
N
N
Ni
i
N
N
000
0
lnlimlnlimlimlnln 
We transformed the Infinite Product in an Infinite Series, and we know that a
necessary (but not sufficient) condition for an Infinite Series to converge is
 1lim0lnlim 

j
j
j
j

For simplicity we will define
0lim1 

j
j
jj aa


1j
j
0
0
lim
N
N
Nj
j
N



Infinite Products
Complex Variables

139. 139
SOLO
Lemma 2:
Let a
j
ϵ C be such that |a
j
| < 1. Let . Then 


N
j
jN aQ
1
1:





N
j j
N
j j a
N
a
eQe
1
1
2
1
Proof:
Since 1 + |a
j
| ≤ e
|aj| 



N
j
j
N
a
Q
N
eaa
1
11
1


On the other hand, since e
x
≤ 1 + 2 x for 0 ≤ x ≤ 1,
    
NN
aa
a
Qaaeee
N
j
j
N
j
jNN
j j





2/212/21
1
22
2
2
1
11
1
1

q.e.d.
Proof: Suppose . Then, by the previous Lemma, Q
N
≤ e
M
, for
all N. Since Q
1
≤Q
2
≤ …., the sequence of “partial products” {QN} converges.
Conversely, if the Infinit Product converges to Q, then Q ≥ 1 and
for all N. Then converges.
 Ma
j
j

1
1


1j j
a
Qa
N
j j ln2
1


Lemma 3:
Let a
j ϵ C be such that |a
j| < 1. Then converges if and only if
converges.
 



1
1
j
ja


1j ja
q.e.d.
Infinite Products
Complex Variables

140. 140
SOLO
Proof: Since the product converges, then |a
j
| → 1, so that a
j
≠ 0
for j ≥ j
0. Let assume j
0 = 1, and define
 



1
1
j
ja
  


N
j
jN
N
j
jN aQandaP
11
1:,1:
Note that for a suitable choice of indexes a
jk

 

N
n
n
k
j
N
j
jN
k
aaP
1 11
11:
Then 11
1 11 1
 
  
N
N
n
n
k
j
N
n
n
k
jN QaaP
kk
and for N, M > 1, N > M
   
  
MN
n
Mj
jM
n
Mj
j
M
j
j
NM
N
j
M
j
jjMN
QQaQ
aaaaPP



 



 
11
11111
1
111 1
Hence, {P
N
} is a Cauchy Sequence, since {Q
N
} is, and it converges.
Infinite Products
Complex Variables
Lemma 4:
If the infinite product converges, then also
converges. Hence if the series converges, also
converges.
 



1
1
j
ja 



1
1
j
j
a




1
1
j
j
a

1j
ja

141. 141
SOLO
Proof (continue – 1):
We need to prove that {P
N
} does not converge to zero. By Lemma 2
 
2
3
1 




N
Mj jaN
Mj
j
ea
for M ≥ j0, and N > M. Then using
  
2
1
1
2
3
1111  

N
Mj
j
N
Mj
j
aa
for M ≥ j
0, and N > M. Hence 
2
1
1 

N
Mj
j
a
so that
  01
2
1
11limlim
0
11
 


j
j
j
N
Mj
j
M
j
j
j
N
j
aaaP
q.e.d.
11 
NN QP
Infinite Products
Complex Variables
Lemma 4:
If the infinite product converges, then also
converges. Hence if the series converges, also
converges.
 



1
1
j
ja 



1
1
j
j
a




1
1
j
j
a

1j
ja

142. 142
SOLO
The Mittag-Leffler and Weierstrass , Hadamard Theorems
Magnus Gösta Mittag-Leffler
1846 - 1927
Karl Theodor
Wilhelm
Weierstrass
(1815 – 11897)
We want to answer the following questions:
• Can we find f ϵ M (C) so that f has poles exactly a
prescribed sequence {z
n
} that does not cluster in C, and
such that f has prescribed principal parts (residiu) at
these poles (this refers to fixing the entire portion of the
Laurent Series with negative powers at each pole)?
A positive answer to this question was given by Mittag-Leffler
• Can we find f ϵ H (C) so that f has zeros exactly at a
given sequence {z
n
} ?
A positive answer to this question was given by Weierstrass
and improved by Hadamard
Infinite Products
Complex Variables
Jacques Salomon
Hadamard
(

1865– 1963
)

143. 143
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Mittag-Leffler’s Expansion Theorem
Magnus Gösta Mittag-Leffler
1846 - 1927
 













1
11
0
n nn
n
aaz
afResfzf
Suppose that the only singularities of f (z) in the z-plane are the
simple poles a
1
, a
2
,…, arranged in order of increasing absolute
values. The respective residues of f are Res { f (a
1
)}, Res { f (a
2
)}, …
C
x
y
R
N
1
a
na
C
N

Proof:
Assume ξ is not a pole of f (z), then has simple poles
at a1, a2,…, and ξ.

z
zf
Residue of at a
n
, n = 1,2,… is

z
zf
 
 
 




n
n
n
az
a
afRes
z
zf
az
n
lim
Residue of at ξ is

z
zf 



 f
z
zf
z
n
az




lim
Let take a circle C
N
at the origin with a radius R
N
→ ∞
By the Residue Theorem







N
N
Cinn n
n
C
a
afRes
fdz
z
zf



Assume f (z) is analytic at z = 0, then





N
N
Cinn n
n
C
a
afRes
fdz
z
zf
0

144. 144
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Mittag-Leffler’s Expansion Theorem (continue – 1)
C
x
y
R
N
1
a
n
a
C
N

Proof (continue – 1):
Let take a circle C
N at the origin with a radius R
N → ∞







N
N
Cinn n
n
C
a
afRes
fdz
z
zf
i 

2
1





N
N
Cinn n
n
C
a
afRes
fdz
z
zf
i
0
2
1

 




























NN
N
CC
Cinn nn
n
dz
zz
zf
i
dz
zz
zf
i
aa
afResff





2
11
2
1
11
0
Since | z-ξ | ≥ | z | - | ξ |=R
N
- | ξ | for z on CN, we have if | f(z) | ≤ M

  
0
2
limlim 








NN
N
R
C
R
RR
RM
dz
zz
zf
N
N
N


0lim 

N
N
C
R
dz
zz
zf

 













1
11
0
n nn
n
aaz
afResfzf
Therefore using this result and ξ → z, we obtain
q.e.d.

145. 145
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Mittag-Leffler’s Expansion Theorem (continue – 1)
Example: Expand 1/sinz
Define
zz
zf
1
sin
1
: 

0
cossincos
sin
lim
cossin
cos1
lim
sin
sin
lim
1
sin
1
lim0
0
'
0
'
00

















zzzz
z
zzz
z
zz
zz
zz
f
z
HopitalL
z
HopitalL
zz
f (z) has Simple Poles at n π, n=±1, ±2,… with Residue
  

n
nz
HopitalL
nznznz
z
z
nz
zz
nzzf
1
cos
1
lim
sin
lim
1
sin
1
limRes
'























 


  




















































1
222
11
1
0
1
12
11
1
11
1
11
Res0
1
sin
1
n
n
n
n
n
n
n nn
n
nz
z
nnznnz
aaz
aff
zz
zf


146. 146
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Generalization of Mittag-Leffler’s Expansion Theorem
q.e.d.
Suppose that the only singularities of f (z) in the z-plane are the
poles a
1
, a
2
,…, arranged in order of increasing absolute values, and
having Higher Order then One. The respective residues of f are
Res { f (a
1
)}, Res { f (a
2
)}, … Suppose that exists a Positive Integer p
such that for |z| = R
N
|f (z)| < R
N
p+1

and the poles a
1
, a
2
,…, a
n
are all inside the Circle of Radius R
N

around the origin (|a
1|≤ |a
2|≤…≤ |a
n | < R
N). Then





 



 






















p
i j
p
j
p
jjj
j
ii
j j
p
j
p
i
p
p
a
z
a
z
aaz
af
i
zf
aza
z
aff
p
z
f
z
fzf
1 1
12
1
1
1
1
11
Res
!
0
Res0
!
0
!1
0


Proof: Start with the Integral



































Nj
N
Cina j
p
j
j
p
w
p
zw
C
p
zaa
af
zww
wf
zww
wf
dw
zww
wf
i
I
11
0
1
1
Res
ResRes
2
1

C
x
y
R
N
1
a
n
a
C
N

Return to Infinite Product

147. 147
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Generalization of Mittag-Leffler’s Expansion Theorem
Proof (continue – 1):
but






111
limRes




















pp
zw
p
zw z
zf
zww
wf
zw
zww
wf
C
x
y
R
N
1a
na
C
N





 

i
i
ip
pp
i
w
p
p
p
p
w
p
w
wd
wfd
zwwd
d
ipi
p
p
zww
wf
w
wd
d
pzww
wf


































1
!!
!
!
1
lim
!
1
limRes
1
0
0
1
1
0
1
0



1
1
!11














ip
ip
ip
p
zw
ip
zwwd
d

 



























p
i
ip
i
p
i
i
i
ip
ip
w
p
w
zi
f
wd
wfd
zw
ip
ipi
p
pzww
wf
0
1
0
1
0
1
0
!
0
!1
!!
!
!
1
limRes
Therefore
Leibnitz Formula for Repeated
Differentiation of a Product

148. 148
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Generalization of Mittag-Leffler’s Expansion Theorem
Proof (continue – 2):
but


 







Nj
Cina j
p
j
j
p
i
ip
i
p
zaa
af
zi
f
z
zf
1
0
11
Res
!
0
C
x
y
R
N
1a
na
C
N

Therefore



































Nj
N
Cina j
p
j
j
p
w
p
zw
C
p
zaa
af
zww
wf
zww
wf
dw
zww
wf
i
I
11
0
1
1
Res
ResRes
2
1


  


0max
2
2
1
2
1
1
max
11


 





N
p
N
N
C
N
N
Rn
RwfC
N
p
N
N
C
p
wf
zRR
R
dw
zww
wf
i
I




 

0
Res
!
0
1
1
0
11


 





j j
p
j
j
p
i
ip
i
p
zaa
af
zi
f
z
zf


 






 

1
1
1
0
Res
!
0
j j
p
j
p
j
p
i
ii
aza
zaf
i
zf
zf

149. 149
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Generalization of Mittag-Leffler’s Expansion Theorem
We can see that for p = 0 we get
C
x
y
R
N
1a
na
C
N



 
 

















11
11
Res0
Res
0
n nn
n
n nn
n
aaz
aff
aza
zaf
fzf
We recovered the Mittag-Leffler’s Expansion Theorem


 






 

1
1
1
0
Res
!
0
j j
p
j
p
j
p
i
ii
aza
zaf
i
zf
zf





















 1
12
1
1
1
11
Res
Res
j
p
j
p
jjj
j
j j
p
j
p
j
a
z
a
z
aaz
af
aza
zaf

q.e.d.
Proof (continue – 3):

150. 150
SOLO
Start with some introductory results:
Theorem
Let f (z) be entire holomorphic (analytic for all z ϵ C) and f (z) ≠ 0
everywhere. There is an entire function g (s) for which f = e
g
.
Corollary
If f (z) is entire Holomorphic (analytic) with finitely many zeros {a
i
≠0}(with
multiplicity) and m zeros at z=0, then there exists an entire g (z) such that


 
n
zgm
azezzf /1
Proof:
Since is entire with no zeros we can apply the previous
Theorem
  
n
m
azzzf /1/
q.e.d.
The Weierstrass Factorization Theorem
Karl Theodor Wilhelm
Weierstrass
(1815 – 11897)
Infinite Products
Complex Variables


zf
zf
zf
zd
d '
ln 
Proof :
Since f (z) ≠ 0 and entire, f’ (s) is also entire, and so is f’(z)/ f (z), therefore



entireiszf
zd
d
zf
zf
zg
zd
d
ln
'
: 
and taking g (0)= 1 we obtain
zg
ezf
q.e.d.

151. 151
SOLO
The Weierstrass Factorization Theorem
Definition
We define the Weierstrass Elementary Factors as












,2,11
01
,
2
2
nez
nz
nzE
n
zz
z
n
Lemma
For |z| ≤ 1, |1 – E (z,n)| ≤ |z|
n+1
.
Proof:
The case n = 0 is trivial. Let n ≥ 1. Let differentiate E (z,n)
   
n
zz
z
nn
zz
z
nn
zz
z
n
zz
z
nn
zz
z
nnnnn
ezezeezzzenzE
zd
d 





222212
22222
111,
By developing in a Taylor series
 0,
2
2
 



k
nk
k
k
n
zz
z
n
bzbeznzE
zd
d
n

 






0
1
0
1,,
k
k
k
k
k
k zaknzE
sd
d
zanzE
















 

,2,10
0
1,0
21
0
j
jn
b
a
aaa
nEa
jn
jn
n
Karl Theodor Wilhelm
Weierstrass
(1815 – 11897)
Inspired by the fact that
 



321
1
ln&11
32
1
1
ln zz
z
z
ez
z
w have the following
Infinite Products
Complex Variables

152. 152
SOLO
The Weierstrass Factorization Theorem
Definition
We define the Weierstrass Elementary Factors as












,2,11
01
,
2
2
nez
nz
nzE
n
zz
z
n
Lemma
For |z| ≤ 1, |1 – E (z,n)| ≤ |z|
n+1
.
Proof (continue – 1):
 01,
1



k
nk
k
kazanzE
So for |z| ≤ 1

 

1
0
1
1
1
1
1
1
1
11
1
1
1
1
,11
,1






























nn
nk
k
n
nk
k
n
s
nk
nk
k
n
nk
nk
k
n
nk
k
k
znEzazaz
zazzazzanzE

q.e.d.
Infinite Products
Complex Variables

153. 153
SOLO
The Weierstrass Product
Let {z
j
} be a sequence of complex numbers such that lim
j→∞
|z
j
|=+∞. We may
assume that 0 < |z
1
| ≤ |z
2
| ≤… Let {p
j
} be integers. Then the Weierstrass Product
defined as
converges uniformly on every set {|z|≤r}, to a holomorphic entire function F. The
zeros of F are precisely the points {z
j
} counted with the corresponding
multiplicity.






















 
















1
1
2
1
1
2
1,
j
z
z
pz
z
z
z
j
j
j
j
j
p
jjjj
e
z
z
p
z
z
E

Proof:
Let r > 0 be fixed. Let j
0
be such that |z
j
| > r for j ≥ j
0
. Thus,
11
1,


















j
j
p
j
p
j
j
j z
r
z
z
p
z
z
E
By the hypothesis on the p
j’s, 























00
1
1,
jj
p
j
jj
j
j
j
z
r
p
z
z
E
By Weierstrass M (Majorant Test) it follows that
converges uniformly on {|z| ≤ r}, for any r > 0. Then exist C > 0 such that
 



0
1,/
jj
jjpzzE
C
jj
p
z
z
Ep
z
z
E
jj
j
j
eeeCp
z
z
E
j
j
jj j
j





















 












0
0
0
1,1,
1,
C
jj
p
z
z
E
jj j
j
jj j
j
jj
p
z
z
E
eep
z
z
Ep
z
z
Ee
j
j
Taylor
j
j
 

















 




























0
00
0
1,,
1,1,
Infinite Products
Complex Variables

154. 154
SOLO
Genus of the Canonical Product
Infinite Products
Complex Variables
Let {z
j
} be a sequence of nonzero complex numbers having finite exponent of
convergence. Then the Weierstrass Product
is called Canonical Weierstrass Product, and the Smallest Integer p such that
is called the Genus of the Canonical Product..






















 
















1
1
2
1
1
2
1,
j
z
z
pz
z
z
z
j
j
j
j
j
p
jjjj
e
z
z
p
z
z
E



 1 1
1
min
j p
j
p
z

155. 155
SOLO
The Weierstrass Factorization Theorem
Lemma
Let {z
j
} be a sequence of complex numbers such that lim
j→∞
|z
j
|=+∞. Then there
exists an entire function F whose zeros are precisely the {z
j
}, counting multiplicity.
This function is
This is a Generalization of the Fundamental Theorem of Algebra


 








1
1,:
kj
j
k
j
z
z
EssF
Infinite Products
Complex Variables

156. 156
SOLO
The Hadamard Factorization Theorem
Examples:
(1)Polynomials have Order 0.
Let N be the degree of p (z)
for all ε > 0 and a suitable constant C
ε .
  raseCrCazazazp
rNn
n

01
(2) The exponential e
x
has order 1, and more generally, have order n
and no smaller power of r would suffice.
(3) sin z, cos z, sinh z, cosh z have order 1.
(4) exp {exp z} has infinite order.
n
n
nn
rzzr
eeee 
Re
n
x
e
Infinite Products
Complex Variables
Jacques Salomon
Hadamard
(

1865– 1963
)
Definition: Order of an Entire Function
An Entire (analytic for all z ϵ C) Function f is said to be of Order ρ, 0 ≤ ρ ≤ +∞, if
meaning that










rasezf
r
rz


 Osupinf:
0
 CzallforeAzfBAexists
zB




 :0,inf:
0

157. 157
SOLO
The Hadamard Factorization Theorem
Relation berween Order ρ and Integer Genus p of an Entire Function
Infinite Products
Complex Variables
Jacques Salomon
Hadamard
(

1865– 1963
)
Paul Garrett, “Weirstrass and Hadamard Products”, March 17, 2012,
http://www.math.umn.edu/~garrett/
If {z
j
} has finite Order of Convergence ρ
1
, then its Genus p is such that:
- if ρ
1
is not integer, p < ρ
1
< p+1;
- if ρ
1
is an integer, then p = ρ
1
if diverges, and p=ρ
1
-1 if
converges.
 

1
1
/1
j
j
z

 

1
1
/1
j
j
z


158. 158
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Expansion of an Integral Function as an Infinite Product
An Integral Function is a function which is Analytic for all finite values of z.
For example e
z
, sin z, cos z are Integral Functions. An Integral Function may be
regarded as a generalization of a Polynomial.
Let f (z) be an Integral Function (no Poles) with Simple/Non-simple Zeros
at a
1
, a
2
,…,a
n
,.., arranged in increasing order (|a
1
|≤ |a
2
|≤…≤|a
n
|≤…. ).
Suppose that exists a Positive Integer p such that for |z| = R
N
|f (z)| < R
N
p+1

and the Zeros a
1
, a
2
,…, a
n
are all inside the Circle of Radius R
N

around the origin (|a
1|≤ |a
2|≤…≤ |a
n | < R
N).
Then f (z) can be expanded as an Infinite Product (Hadamard):


pi
i
f
f
zd
d
c
e
a
z
efzf
z
i
i
i
j
a
z
pa
z
a
z
j
zczc
p
j
p
jj
p
p
,,1,0
!1
:
10
0
1
1
1
1
1
2
1
1
1
2
2
1
11



































C
x
y
R
N
1
a
n
a
C
N

Note:
1.The minimum p for which |f(z)|<R
N
p+1
is called the Order of f(z)
2.If f(z) has no poles or zeros then the previous relation reduces to

1
11
0



p
pzczc
efzf

The Hadamard Factorization Theorem
Return to Gamma F.
Return to Zeta F.

159. 159
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Expansion of an Integral Function as an Infinite Product
Proof:
C
x
y
R
N
1
a
na
C
N

Let compute: 


zf
zf
zf
zd
d
1
ln 
 
 



1limlimRes
1
21
'
11










 




















f
fazf
f
faz
f
f j
az
HopitalL
j
az
az
jj
j
Define


pi
i
f
f
zd
d
c
z
i
i
i ,,1,0
!1
:
0
1
1 
































1
12
0
1
1
11
1
j
p
j
p
jjj
p
i
i
i
a
z
a
z
aaz
zic
zf
zf

All Zeros of f (z) (a1, a2,…,a
n
,..) are Simple Poles of f
(1)(
z)/f(z), therefore we
can apply the previous result and write:





































1
12
1
0
0
1
1
11
Res
!
j
p
j
p
jjjaz
p
i
i
z
i
i
a
z
a
z
aazf
f
i
z
f
f
zd
d
zf
zf
j

The Hadamard Factorization Theorem

160. 160
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Expansion of an Integral Function as an Infinite Product
Proof (continue – 1):
C
x
y
R
N
1
a
na
C
N

Integrating from 0 to z along a path not passing through any of a
j
, we obtain































1
1
1
2
2
0
1
1
1
1
2
1
ln
0
ln
j
p
j
p
jjj
j
p
i
i
i
a
z
pa
z
a
z
a
az
zc
f
zf

The values of the logarithms will depend on the path chosen, but when we take
exponentials all the ambiguities disappear,




















1
12
0
1
1
11
1
j
p
j
p
jjj
p
i
i
i
a
z
a
z
aaz
zic
zf
zf








 













1
1
1
2
1
1
1
2
2
0
1
1
1
0
j
a
z
pa
z
a
z
j
zc p
j
p
jj
p
i
i
i
e
a
z
e
f
zf

q.e.d.
If |f(z)| < R
N
p+1
it will be true for all q > p. If we choose the ρ = min p for which
the inequality holds, then we obtain the Hadamard’s Factorization .
The Hadamard Factorization Theorem

161. 161
SOLO
Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Example: Expand sinz/z
Define 
z
z
zf
sin
:
 1
1
cos
lim
sin
lim0
0
'
0


z
z
z
f
z
HopitalL
z
f (z) has Simple Zeros at n π, n=±1, ±2,…
Expansion of an Integral Function as an Infinite Product




































1
22
2
11
111
sin
0
nnn n
z
n
z
n
z
z
z
f
zf

 01
sin 0
 pR
z
z
zf
N
Leonhard Euler
(1707 – 1783
)
We recovered the Euler Product Formula 1735

162. 162
SOLO
Analytic continuation (sometimes called simply "continuation") provides a way of
extending the domain over which a complex function is defined. The most common
application is to a complex analytic function determined near a point by a power series
 



0
0
k
k
k
zzazf
Such a power series expansion is in general valid only within its radius of convergence. However, under
fortunate circumstances (that are very fortunately also rather common!), the function will have a power
series expansion that is valid within a larger-than-expected radius of convergence, and this power series
can be used to define the function outside its original domain of definition. This allows, for example, the
natural extension of the definition trigonometric, exponential, logarithmic, power, and hyperbolic
functions from the real line to the entire complex plane . Similarly, analytic continuation can be used to
extend the values of an analytic function across a branch cut in the complex plane.
Analytic Continuation
Analytic continuation of natural logarithm
(imaginary part)
Complex Variables

163. 163
SOLO
Analytic Continuation
Complex Variables

164. 164
SOLO
Complex Variables
Conformal Mapping
Transformations or Mappings
x
y
u
v
r

xd
yd
r

ud
vd
A
B
CD
'A
'B
'C
'D
The set of equations







yxvv
yxuu
,
,
define a general transformation or mapping
between (x,y) plane to (u,v) plane.
If for each point in (x,y) plane there corresponds one and only one point in (u,v)
plane, we say that the transformation is one to one.
vd
v
r
ud
u
r
vdy
v
y
x
v
x
udy
u
y
x
u
x
yvd
v
y
ud
u
y
xvd
v
x
ud
u
x
yydxxdrd
u
r
u
r






































































1111
1111
If is a vector that defines a point A in (x,y) plane, we have: vuryxr ,,

r

The area dx dy of a region A,B,C,D, in (x,y) plane is mapped in the area A’,B’,C’,D’,
du dv in the (u,v) plane. We have

zvdud
u
y
v
x
v
y
u
x
vdudy
v
y
x
v
x
y
u
y
x
u
x
vdud
v
r
u
r
zydxdydxd
y
r
x
r
Sd
yx
11111
1
11




























































If x and y are
differentiable

165. 165
SOLO
Complex Variables
Conformal Mapping
Transformations or Mappings







yxvv
yxuu
,
,
The transformation is one to one if and only if, for distinct points A, B, C, D, in (x,y)
we obtain distinct points A’,B’,C’,D’, in (u,v). For this a necessary (but not sufficient)
condition:
''''
det1det
11
DCBA
ABCD
Sd
v
y
u
y
v
x
u
x
zvdud
v
y
u
y
v
x
u
x
zvdud
u
y
v
x
v
y
u
x
zydxdSd






































Transformation is one to one
00
''''
DCBAABCD SdSd



0det:
,
,












v
y
u
y
v
x
u
x
vu
yxJacobian of the
Transformation
By symmetry (change x,y to u,v) we obtain:
ABCDDCBA
Sd
y
v
x
v
y
u
x
u
Sd









det
''''
1detdet 
















v
y
u
y
v
x
u
x
y
v
x
v
y
u
x
u
one to one
transformation




1
,
,
,
,





vu
yx
yx
vu
x
y
u
v
r

xd
yd
r

ud
vd
A
B
CD
'A
'B
'C
'D

166. 166
SOLO
Complex Variables
Conformal Mapping
Complex Mapping
In the case that the mapping is done by a complex function, i.e.
 yixfzfviuw 
we say that f is a complex mapping.
If f (z) is analytic, then according to Cauchy-Riemann equation:



2222
det
,
,
zd
zfd
y
u
i
x
u
y
u
x
u
x
v
y
u
y
v
x
u
y
v
x
v
y
u
x
u
yx
vu


















































x
v
y
u
y
v
x
u










&
If follows that a complex mapping f (z) is one to one in regions where df/dz ≠ 0.
Points where df/dz = 0 are called critical points.

167. 167
SOLO
Complex Variables
Conformal Mapping
Complex Mapping – Riemann’s Mapping Theorem
In the case that the mapping is done by a complex function, i.e.
 yixfzfviuw 
Georg Friedrich Bernhard
Riemann
1826 - 1866
we have:
x
y
u
v
C
'C
1
R
R' Let C be the boundary of a region R in the z plane,
and C’ a unit circle, centered at the origin of the
w plane, enclosing a region R’.
The Riemann Mapping Theorem states that for each point
in R, there exists a function w = f (z) that performs a
one to one transformation to each point in R’.
Riemann’s Mapping Theorem demonstrates the existence of the
one to one transformation to region R onto R’, but it not provides
this transformation.

168. 168
SOLO
Complex Variables
Conformal Mapping
Complex Mapping (continue – 1)







yxvv
yxuu
,
,
x
y
u
v
r

2
zd
1
zd
r

2
wd
1
wd
A
B
C
'A
'B
'C
 yixfzfviuw 
Consider a point A in (x,y) plane mapped to point
A’ in (u,v) plane
Consider a small displacement from A to B
defined as dz
1
, that is mapped to a displacement
from A’ to B’ defined as dw
1
 











1
1
argarg
11
arg
11
zd
zd
zfd
i
AA
wdi A
ezd
zd
zfd
zd
zd
zfd
ewdwd
Consider also a small displacement from A to C defined as dz
2
, that is mapped to
a displacement from A’ to C’ defined as dw
2
 











2
2
argarg
22
arg
22
zd
zd
zfd
i
AA
wdi
A
ezd
zd
zfd
zd
zd
zfd
ewdwd
We can see that dw ≠ 0 if dz ≠ 0, i.e. a one-to-one transformation, if and only if

0
A
zd
zfd

169. 169
SOLO
Complex Variables
Conformal Mapping
Complex Mapping (continue – 2)







yxvv
yxuu
,
,
x
y
u
v
r

2
zd
1
zd
r

2
wd
1
wd
A
B
C
'A
'B
'C
 yixfzfviuw 
Consider a point A in (x,y) plane mapped to point
A’ in (u,v) plane
 











1
1
argarg
11
arg
11
zd
zd
zfd
i
AA
wdi
A
ezd
zd
zfd
zd
zd
zfd
ewdwd
 











2
2
argarg
22
arg
22
zd
zd
zfd
i
AA
wdi
A
ezd
zd
zfd
zd
zd
zfd
ewdwd
We can see that:
 
 
12
1212
argarg
argargargargargarg
zdzd
zd
zd
zfd
zd
zd
zfd
wdwd
AA



















Consider two small displacements from A to B
And from A to C, defined as dz
1 and dz
2, that are
mapped to displacements from A’ to B’ and from A’ to C’, defined as dw
1
and dw
2
Therefore the angular magnitude and sense between dz
1
to dz
2
is equal to that
between dw
1 to dw
2. Because of this the transformation or mapping is called a
Conformal Mapping.

170. 170
SOLO
Complex Variables
Conformal Mapping
    RzRzzzf 
2/1
22
ln
 
 
 
  
z
Rzz
RzzR
Rzz
Rzz
R
Rzz
w
R
w 2
222
2/1
222
2/1
22
2/1
22
2
2/1
22
2







Define    RzRzzzgw 
2/1
22









w
R
w
2
1
z
2
 
 
   



















w
w
R
w
w
R
wwiww
2
1
wwiww
R
wwiww
2
1
yix
argsinargcosargsinargcos
argsinargcos
argsinargcos
22
2
 


















w
R
ww
2
1
y
w
R
ww
2
1
x
22
argsin&argcos

171. 171
SOLO
Complex Variables
Conformal Mapping
    RzRzzzf 
2/1
22
ln
Define    RzRzzzgw 
2/1
22
 


















w
R
ww
2
1
y
w
R
ww
2
1
x
22
argsin&argcos
From those equations we have:
 
2
2
2
2
2
22
4
1
4
1
argsinargcos
R
w
R
w
w
R
w
w
y
w
x






























4
1
w
R
w
y
w
R
w
x
































2
2
2
2
x
y
warg
wln
 wiwzf argln

172. 172
SOLO
Complex Variables
Conformal Mapping
 2/
tt
eex


012
2

tt
exe
 2/cosh
tt
eet


  
2/1
2
1ln  xxxacosh
http://www.mathworks.com/company/newsletters/news_notes/clevescorner/sum98cleve.html

173. 173
SOLO
Complex Variables
Conformal Mapping
   
2/1
22
ln Rzzzf 
 
 
 
  
z
Rzz
RzzR
Rzz
Rzz
R
Rzz
w
R
w 2
222
2/1
222
2/1
22
2/1
22
2
2/1
22
2







Define   
2/1
22
Rzzzgw 









w
R
w
2
1
z
2
 
 
   



















w
w
R
w
w
R
wwiww
2
1
wwiww
R
wwiww
2
1
yix
argsinargcosargsinargcos
argsinargcos
argsinargcos
22
2
 


















w
R
ww
2
1
y
w
R
ww
2
1
x
22
argsin&argcos

174. 174
SOLO
Complex Variables
Conformal Mapping
   
2/1
22
ln Rzzzf 
Define   
2/1
22
Rzzzgw 
 


















w
R
ww
2
1
y
w
R
ww
2
1
x
22
argsin&argcos
From those equations we have:
 
2
2
2
2
2
22
4
1
4
1
argsinargcos
R
w
R
w
w
R
w
w
y
w
x






























4
1
w
R
w
y
w
R
w
x
































2
2
2
2
x
y
warg
wln
 wiwzf argln

175. 175
SOLO
Complex Variables
Conformal Mapping
http://www.mathworks.com/company/newsletters/news_notes/clevescorner/sum98cleve.html
 2/
tt
eex


012
2

tt
exe
 2/sinh
tt
eet


   
2/1
2
1ln  zzzasinh:zf

176. 176
SOLO
Complex Variables
Conformal Mapping
dz
dz
kviuw


 ln


ku
e
ydx
ydx
dz
dz
/2
22
222






 
  dx
dyx
ydxydx
ydxydx
e
e
k
u
ku
ku
21
1
coth
222
2222
2222
/2
/2














    kudkudykudx /sinh/1/coth/coth
222222

kvikukviku
ee
dz
dz
ee
dz
dz
//// 








dyi
dyx
i
dzdz
dzz
i
dz
dz
dz
dz
dz
dz
dz
dz
i
ee
ee
i
k
u
ctg
kvikvi
kvikvi
2
2222
//
//
































    kvdkvctgdkvctgdyx /sin/1//
222222

kviku
ee
dz
dz
//




177. 177
SOLO
Complex Variables
Conformal Mapping
dz
dz
kviuw


 ln
  kudykudx /sinh//coth
2222

  kvdkvctgdyx /sin//
2222

dd
x
y
  kvdkvctgdyx /sin//
2222

  kudykudx /sinh//coth
2222

1
v
2v
3
v
3u
2
u
1
u
We have two families of orthogonal circles.
All those circle passe through (-d,0) and (d,0)
v
u

178. 178
SOLO
Complex Variables
Conformal Mapping
http://www.mathworks.com/company/newsletters/news_notes/clevescorner/sum98cleve.html
1
1
2
2



w
w
e
e
z
 zze
w
11
2

1
1
tanh
2
2








w
w
ww
ww
e
e
ee
ee
w
 








z
z
zatanhw
1
1
ln
2
1

179. 179
SOLO
Complex Variables
Conformal Mapping
The complex squaring map
(on left half square)
The complex squaring map
(on right half square)
The complex squaring map
(on entire square)

2
zzf
-3/2
-3/2
+3/2
+3/2
x
y
Transform the square under the map
Douglas N. Arnold

180. 180
SOLO
Complex Variables
Conformal Mapping
The complex exponential map

z
ezf
Transform the strip ± i π under the exponential map
Douglas N. Arnold
+i π
x
y
-i π

181. 181
SOLO
Complex Variables
Conformal Mapping
The complex cosine map
Douglas N. Arnold
 zzf sin
-π
-1
+π
+1
x
y
Transform the square under the maps
The complex sine map
 zzf cos

182. 182
SOLO
Complex Variables
Conformal Mapping
Douglas N. Arnold
An important property of analytic functions is that they are conformal maps
everywhere they are defined, except where the derivative vanishes. A conformal map
is one that preserves angles. More precisely, if two curves meet at a point and their
tangents make a certain angle there, then the angle between the images curves under
any analytic function (with non-vanishing derivative) will be the same in both sense
and magnitude


zzf

183. 183
SOLO
Complex Variables
Mobius Transformation
Douglas N. Arnold

 
10
/11
4



 t
zit
itz
zf
t

August Ferdinand Möbius
1790 - 1868

184. 184
SOLO
Complex Variables
Schwarz-Christoffel Mappings
Hermann Amandus
Schwarz
1843 - 1921
Elwin Bruno Cristoffel
1829 - 1900
1

2

3

4

5
6
1
w
2w
3
w
4
w
5w
6w
u
v
x
y
1x
2
x
3x
4
x
5x
6x
A Schwarz – Christoffel transformation is an analytic mapping of
The upper half-plane (x,y) onto a polygon in (u,v) plane.
Let take n points on x axis:
n
xxx  
21
Define the derivative of the mapping as:

11
2
1
1
21








n
n
xzxzxzA
zd
fd
zd
wd

or
  BdzxzxzxzAzfw
n
n 

 11
2
1
1
21







where A and B are complex constants.

185. 185
SOLO
Complex Variables
Schwarz-Christoffel Mappings
1
2

3

4
5

6

1w
2w
3
w
4w
5
w
6
w
u
v
x
y
1x
2x
3x
4x
5x
6x

11
2
1
1
21








n
nxzxzxzA
zd
fd
zd
wd

Since for x
i-1
< x < x
i
the slope of d w/ d z is constant, i.e. the real axis is mapped in
straight lines.
We can see that for x > x
n
: A
zd
wd
argarg 








  
1
1
1
1 arg1arg1argarg






























ni
xzxzA
zd
wd
ni
For x
i-1 < x < x
i:
For x > x
i
:   
1
1
1
1
1
arg1arg1argarg
1






















 








ni
xzxzA
zd
wd
ni

(1) Any three of the points can be chosen at will.
n
xxx  
21
(2) The constants A and B determine the size, orientation and position of the polygon.
(3) If we choose x
n
at infinity, the last term that includes x
n
is not present.
(4) Infinite open polygons are limiting cases of closed polygons.

186. 186
SOLO
Complex Variables
Schwarz-Christoffel Mappings
Douglas N. Arnold
According to the Riemann mapping theorem, there exists a conformal map from the
unit disk to any simply connected planar region (except the whole plane). However,
finding such a map for a specific region is generally difficult. An important special
case where a formula is known is when the target region is polygonal. In that case we
have the Schwarz-Christoffel formula, written as
 



zn
j
j
dzcfzf
j
01
1
0 

Here the polygon has n vertices, the interior angles at the vertices
are , , in counterclockwise order, and c is a complex constant. The
numbers , , are the pre-images of the polygon's vertices, or prevertices,
which lie in order on the unit circle.
n ,,
1
nzz,,
1
The first animation illustrates the effect
of the prevertices. The prevertices start
in a random configuration, and the
resulting image polygon is shown. Then
the prevertices are moved (linearly in
argument) into a configuration leading
to a symmetric "X." Notice how the
angles remain fixed, but the side lengths
vary nonlinearly into the final
configuration.

187. 187
SOLO
Complex Variables
Schwarz-Christoffel Mappings
Douglas N. Arnold
 



zn
j
j
dzcfzf
j
01
1
0 

Here the polygon has n vertices, the interior angles at the vertices
are , , in counterclockwise order, and c is a complex constant. The
numbers , , are the pre-images of the polygon's vertices, or prevertices,
which lie in order on the unit circle.
n
 ,,
1

nzz,,
1
A variation on the first animation is to leave the
prevertices fixed and vary the angles assigned to
them. Here we "square" the ends of the X into right
angles. The color of a (pre)vertex indicates its
distance from being a right angle.
The last sequence The color indicates the radius of a
point's image in the disk. Notice how the arms of the X
originate from points quite close to the boundary of
the disk.

188. 188
SOLO
Complex Variables
Applications of Complex Analysis
Douglas N. Arnold
Gamma Function
Bernoulli Numbers
Fourier Transform
Laplace Transform
Z Transform
Mellin Transform
Hilbert Transform
Zeta Function

189. 189
SOLO
Primes





t
t
t
z
td
e
t
z
0
1
Proof:
Gamma Function
0& xyixz









t
t
t
zt
t
t
zt
t
t
z
td
e
t
td
e
t
td
e
t
1
11
0
1
0
1
For the first part:
x
t
xx
t
x
tdttd
e
t
td
e
t
x
t
t
t
x
t
t
x
e
t
t
t
yixt
t
t
z t
1
lim
111
0
1
0
1
0
1
1
1
0
11
0
1














The first integral converges for any x ≥ δ > 0.
For the second integral, using integration by parts:
































t
t
t
x
e
t
t
tx
edv
tu
t
t
t
x
e
t
t
tx
edv
tu
t
t
t
xt
t
t
yixt
t
t
z
td
e
t
xxetx
e
td
e
t
xettd
e
t
td
e
t
td
e
t
t
x
t
x
1
3
/1
1
2
1
2
/1
1
1
1
1
1
1
1
1
211
1
1
2
1


Euler’s Second Integral
Gamma integral is defined, and
converges uniformly for x > 0.

190. 190
SOLO
Primes





t
t
t
z
td
e
t
z
0
1
Proof (continue):
Gamma Function
0& xyixz
For the second integral, using integration by parts:
































t
t
t
x
e
t
t
tx
edv
tu
t
t
t
x
e
t
t
tx
edv
tu
t
t
t
xt
t
t
yixt
t
t
z
td
e
t
xxetx
e
td
e
t
xettd
e
t
td
e
t
td
e
t
t
x
t
x
1
3
/1
1
2
1
2
/1
1
1
1
1
1
1
1
1
211
1
1
2
1


After [x] (the integer defined such that x-[x] < 1) such integration the power of t in
the integrand becomes x-[x]-1 < 0. and we have:
  
 
    





t
t
t
t
t
txx
td
e
xxxxtd
et
xxxx
11
1
1
21
1
21 
Therefore the Gamma integral is defined, and converges uniformly for x > 0.
Gamma integral is defined, and
converges uniformly for x > 0.
q.e.d.

191. 191
SOLO





t
t
t
z
td
e
t
z
0
1
Proof:
Gamma Function
0& xyixz
zzz 1

  zztdetztdtzeettdetz
t
t
tz
t
t
ud
z
v
t
v
t
u
z
dtedvtu
partsby
t
t
tz
tz
 













0
1
0
1
0
,
nintegratio
0
01

Properties of Gamma Function:
1
Note that for the evaluation of Gamma Function for a Positive Real Number
we need to know only the value of Γ (x) for 0 < x < 1
  xxxnxnxnx  121 


  121 


nxnxxx
nx
x

For x < 0 with –n < x < -n+1 or 0 < x+n < 1, we define
We can see that for x = 0 or a negative integer the
denominator of the right side is zero, and so Γ (x) is
undefined (goes to infinity)
Gamma Function
 ,2,1,0!1  nnn

192. 192
SOLO
Primes





t
t
t
z
td
e
t
z
0
1
Proof:
Gamma Function


!1
1
Residue
1
1 



 n
z
n
nz
Residues of Gamma Function at x = 0,-1, -2,---,-n:..,


  121 


nxnxxx
nx
x

q.e.d.
  

  


  





,2,1
!1
1
121
1
121
1limResidue
1
1
11












n
nnn
nxnxxx
nx
nxx
n
nxnx

193. 193
SOLO
Primes





t
t
t
z
td
e
t
z
0
1
Gamma Function
Absolute value |Γ (z)|
Real value ReΓ (z)
Imaginary value ImΓ (z)

194. 194
SOLO
Primes





t
t
t
z
td
e
t
z
0
1
Gamma Function
zzz 1
Let compute
 11
0
0







t
t
t
t
etde
Therefore for any n positive integer:
 !1122112111  nnnnnnnnn 
Properties of Gamma Function :
1
2
q.e.d.

195. 195
SOLO
Primes
Second definition identical to First
    bayxallyfxfyxf ,,1,011  
xa
b
yyx 1
 yxf  1
yfxf  1
Convex Function:
A Function f (x) is called Convex in an interval (a,b) if for every x,y ϵ (a,b) we have
A Function f (x), defined for x > 0, is called Convex, if the corresponding function


y
xfyxf
y


defined for all y > -x, y ≠ 0, is monotonic Increasing throughout the range of
definition.
x yxy
yxf
xf
If 0 < x
1
< x < x
2
, are given by choosing y
1
= x
1
– x < 0, y
2
= x
2
– x > 0, we express
the condition of convexity as




xx
xfxf
y
xx
xfxf
y






2
2
2
1
1
1

   xxxfxfxxxfxf 
1221

 
 


 

 






1
12
1
2
12
2
1
xx
xx
xf
xx
xx
xfxf
One other equivalent definition:

196. 196
SOLO
Primes
   1,0ln1ln1ln   yfxfyxf
Logarithmic Convex Function :
A Function f (x)>0 is called logarithmic-convex or simply log-convex if ln (f (x) )
is convex or
This is equivalent to   




1
ln1ln yfxfyxf
Since the logarithm is a momotonic increasing function we obtain
 

 yxyfxfyxf 

,1,01
1



197. 197
SOLO
Primes





t
t
t
z
td
e
t
z
0
1
Proof :
Gamma Function
0& xyixz
   1,0ln1ln1ln   baba
Properties of Gamma Function :
3
Gamma is a
Log Convex
Function
 

  






































1
1
0
1
0
1
0
1
11
0
11
1
badtetdtet
dtetetdtetba
tbta
InequalityHolder
tbtatba
q.e.d.

198. 198
SOLO
Primes





t
t
t
z
td
e
t
z
0
1
Proof :
Gamma Function
Other Gamma Function Definitios:

nxxx
nn
x
x
n 

 1
!
limGauss’ Formula
Since the Gamma Function is monotonically increasing the logarithm of Gamma
Function is also monotonic increasing and for 0 < x < 1 and any n > 2 we have
 
nnx
nnx

 lnln
  


  























!1
!
ln
!2
!1
ln
1
!1ln!ln!1lnln
1
!1ln!2ln
n
n
n
n
nn
x
nnxnn



n
x
n
nx
n ln
!1
ln
1ln 



x1 1
yln
0
  
 
1
1
ln1ln




x
nn
nn  
nn
nn
x




1
ln1ln
1
Carl Friedrich Gauss
(1777 – 1855)

199. 199
SOLO
Primes





t
t
t
z
td
e
t
z
0
1
Proof (continue - 1) :
Gamma Function
Other Gamma Function Definitios:
Since the Gamma Function is monotonically increasing the logarithm of Gamma
Function is also monotonic increasing and for 0 < x < 1 and any n > 2 we have



n
x
n
nx
n ln
!1
ln
1ln 


 


xx
n
n
nx
n ln
!1
ln1ln 



10x
!1!11  nnnxnn
xx
Use  xxxnxnxnx 



0
121

  


  xxnxnx
nn
x
xxnxnx
nn
xx
121
!1
121
!11







nxxx
nn
x
x
n 

 1
!
limGauss’ Formula
Euler 1729
Gauss 1811

200. 200
SOLO
Primes





t
t
t
z
td
e
t
z
0
1
Proof (continue - 2) :
Gamma Function
Other Gamma Function Definitios:

  


  xxnxnx
nn
x
xxnxnx
nn
xx
121
!1
121
!11






 


 xxnxnx
nn
x
xxnxnx
nn
xx
11
!1
11
!



 
Take the limit n → ∞
 

 xxnxnx
nn
n
x
xxnxnx
nn
x
n
x
n
x
n 11
!
lim
1
1lim
11
!
lim
1









 



 
1,0
11
!
lim 



x
xxnxnx
nn
x
x
n 
Substitute n+1 for n

nxxx
nn
x
x
n 

 1
!
limGauss’ Formula

201. 201
SOLO
Primes





t
t
t
z
td
e
t
z
0
1
Let substitute x + 1 for x
Gamma Function
Other Gamma Function Definitios:

 

1,0
11
!
lim 




x
xxnxnx
nn
x
x
x
n
n


q.e.d

nxxx
nn
x
x
n 

 1
!
limGauss’ Formula
Proof (continue - 3) :

 
  

 1,0
11
!
lim
1
lim
11
!
lim1
1
1









xxx
xxnxnx
nn
nx
n
x
xnxnx
nn
x
x
x
nn
x
n




The right side is defined for 0 < x <1. The left side extend the definition for
(1 , 2). Therefore the result is true for all x , but 0 and negative integers.

202. 202
SOLO
Primes





t
t
t
z
td
e
t
z
0
1
Gamma Function
Other Gamma Function Definitios:
Start from Gauss Formula xx
n
n


lim
q.e.d

constantMascheroni-Euler57721566.0ln
1
2
1
1lim
1
1














n
n
k
x
e
x
e
x
n
k
k
x
x


Weierstrass’ Factorization Formula for Gamma Function
Proof :

 



















































n
x
n
xx
x
eee
e
x
x
n
x
n
x
n
xxnxnx
nn
x
n
xxx
n
nx
xx
n
1
1
1
1
1
1
1
1
11
11
!
:
21
1
2
1
1ln





  
















11
1
2
1
1ln
11
1
limlim
k
k
x
xn
k
k
x
n
nx
n
n
n
k
x
e
x
e
k
x
e
x
exx


Karl Theodor Wilhelm
Weierstrass
(1815 – 11897)

203. 203
SOLO
Primes





t
t
t
z
td
e
t
z
0
1
Gamma Function
Other Gamma Function Definitios:
Weierstrass’ Factorization Formula for Gamma Function (continue)
Karl Theodor Wilhelm
Weierstrass
(

1815 – 11897)














1
1
1
k
k
z
z
e
k
z
ez
z

Γ (z) has Poles at z
k = - k, k=0,1,2,…, and no Zeros therefore
1/ Γ (z) has Zeros at z
k
= - k, k=0,1,2,…, and no Poles, and
From previous development we obtain Weierestrass
Factorization
Return to ζ (z)

204. 204
SOLO





t
t
t
z
td
e
t
z
0
1
Gamma Function
Gamma integral is defined, and
converges uniformly for x > 0.
Differentiation of Gamma Function:
q.e.d





 





0,2
!11'
ln
0
1'''
ln
constantMascheroni-Euler57721566.0
111'
ln
1
1
1
1
22
2
2
2
1



































xn
kx
n
x
x
xd
d
x
xd
d
kxx
xxx
x
xd
d
kxkxx
x
x
xd
d
k
n
n
n
n
n
n
k
k

Proof :
Start from Weierstrass Formula





1
1
k
k
x
x
k
x
e
x
e
x

 











11
1lnlnln
kk k
x
k
x
xxx  






11
1
1
11
ln
kk
k
x
k
kx
x
xd
d


 
0
111111
ln
0
2
1
22
1
2
2




















 





 kkk kxkxxkxkxxd
d
x
xd
d

















0
1
1
!11'
ln
k
n
n
n
n
n
n
kx
n
x
x
xd
d
x
xd
d
Gamma Function
We can see that 


 















1
1
1
1 1
11
lim
1
1
1
1'
1ln
n
n
k
n kk
x
xd
d

205. 205
SOLO
Primes





t
t
t
z
td
e
t
z
0
1
Gamma Function
Other Gamma Function Definitios:
 















1
1
1
1
1
k
k
z
z
e
k
z
e
zzz

Return to ζ (z)
Hadamard Infinite Product Expansion of Gamma Function
 


















1
1
1
2
1
1
1
2
2
1
11
10
j
a
z
pa
z
a
z
j
zczc
p
j
p
jj
p
p
e
a
z
efzf


Since 1/z Γ (z)=1/Γ (1+z) has Zeros at z
k
= - k, k=1,2,…, and no Poles we can use
the Hadamard Infinite Product Expansion


pi
i
f
f
zd
d
c
z
i
i
i ,,1,0
!1
:
0
1
1 












Gamma Function Γ (1+z) has Order p=0, and

















1
1'
:
0
1
1
z
f
f
c
 1101  fzzfDefine
We recovered the
Weierstrass Formula
using Hadamard
Expansion

206. 206
SOLO
Primes
 




1
0
11
1,
s
s
zy
sdsszyBBeta Function
Beta Function is related to Gamma Function:
 









u
u
uy
duudt
ut
t
t
ty
udeutdety
0
12
2
0
1
2
2
2


zy
zy
zyB


,
Proof:
In the same way:





v
v
vz
vdevz
0
12
2
2

 







u
u
v
v
vuuzy
vdudevuzy
00
1212
22
4
Use polar coordinates:








drdrdrd
r
r
drd
vrv
uru
vdud
rv
ru











cossin
sincos
//
//
sin
cos


  


  




































2/
0
1212
0
12
0
2/
0
121212
sincos22
sincos4
2
2






drder
drderzy
zy
zy
r
r
rzy
r
r
rzyzy

Euler’s First Integral

207. 207
SOLO
Primes
 




1
0
11
1,
s
s
zy
sdsszyBBeta Function Euler’s First Integral
Beta Function is related to Gamma Function:


zy
zy
zyB


,
Proof (continue):
   








 



2/
0
1212
sincos2


 dzyzy
zy
Change variables in the integral using
 dsds cossin2sin
2

    zyBsdssd
s
s
yzzy
,1sincos2
1
0
11
2/
0
1212










zyBzyzy ,
Therefore q.e.d.
Use z→y and y → 1 - z
  

 






























u
u
zu
u
z
z
z
u
u
s
u
ud
sd
s
s
zz
ud
u
u
u
ud
u
u
u
u
dssszzBzz
0
1
0
21
1
1
1
1
0
1
111
1
1
11,11
2 q.e.d.

208. 208
SOLO
Primes
Proof


yzBzyzyBzyyz
yzBzyB
,,
,,


Use y → 1 - z
  

 






























u
u
zu
u
z
z
z
u
u
s
u
ud
sd
s
s
zz
ud
u
u
u
ud
u
u
u
u
dssszzBzz
0
1
0
21
1
1
1
1
0
1
111
1
1
11,11
2





t
t
t
z
td
e
t
z
0
1
Gamma Function
Other Gamma Function Properties:

z
zz


sin
1
Euler
Reflection Formula

209. 209
SOLO
Primes
Proof (continue - 1)






u
u
x
ud
u
u
xx
0
1
1
1





t
t
t
z
td
e
t
z
0
1
Gamma Function
Other Gamma Function Properties:
R

R
C
C
1
planeu
uRe
uIm
Replace the path from 0 to ∞ by the Hankel contour H
ε
in the Figure, described by four paths, traveled in
counterclockwise direction :
1 .going counterclockwise above the real axis, (u = |u|)
2 .along the circular path C
R
,
3 .bellow the real axis, (u= |u|e
-2πi
)
4 .along the circular path C
ε
.














 C
yR y
yi
C
yR y
ud
u
u
ud
u
u
eud
u
u
ud
u
u
R
1111
2
Define y = 1 – x, and assume x,y ϵ (0,1)

z
zz


sin
1
Euler
Reflection Formula

210. 210
SOLO
Primes
Proof (continue - 1)






u
u
x
ud
u
u
xx
0
1
1
1





t
t
t
z
td
e
t
z
0
1
Gamma Function
Other Gamma Function Properties:
R

RC

C
1
planeu
uRe
uIm
This path encloses the pole u=-1 of that has the residue
1

u
u
y
yi
eu
y
y
eu
u
u
i
















1
1
Residue
By the Residue Theorem
For z ≠ 0 we have

yzyzyzyy
zeeez


lnlnReln

z
zz


sin
1
Euler
Reflection Formula

yi
y
eu
y
C
yR y
iy
C
yR y
ei
u
u
ui
u
u
izd
z
z
ud
u
u
ezd
z
z
ud
u
u
i
R















































2
1
1lim2
1
Residue2
1111
1
2

211. 211
SOLO
Primes
Proof (continue - 2)





t
t
t
z
td
e
t
z
0
1
Gamma Function
Other Gamma Function Properties:
R

R
C

C
1
planeu
uRe
uIm
yi
C
yR y
iy
C
yR y
eizd
z
z
ud
u
u
ezd
z
z
ud
u
u
R



















2
1111
2
For the second and forth integral we have

0
lnlnReln


zzeeez
yzyzyzyy
z
z
z
z
z
z
yy
y







111
Hence for small ε we have :
and for large R we have :
0
1
2
1
0
1












y
C
y
zd
z
z
0
1
2
1
1







R
y
C
y
R
R
zd
z
z
R

Therefore the integrals on the circular paths are zero for ε→0 and R∞→

z
zz


sin
1
Euler
Reflection Formula

212. 212
SOLO
Primes
Proof (continue - 3)





t
t
t
z
td
e
t
z
0
1
Gamma Function
Other Gamma Function Properties:
R

R
C

C
1
planeu
uRe
uIm
yi
y
iy
y
eiud
u
u
eud
u
u











2
11
0
2
0
We obtain
Multiply both sides by
yi
e

  iud
u
u
ee
y
iyiy


2
1
0





  yee
i
ud
u
u
iyiy
y




sin
2
1
0







Rearranging we obtain
Since both sides of this equation are Holomorphic (analytic) in x ϵ (0,1) we can
extend the result for all analytic parts of z ϵ C (complex plane).

  
1,0
sin1sin11
1
0
1
0
1






 






x
xx
ud
u
u
ud
u
u
xx
u
u
yxy
u
u
x




Substituting y = 1 – x we obtain

z
zz


sin
1
Euler
Reflection Formula

213. 213
SOLO
Primes
Onother Proof





t
t
t
z
td
e
t
z
0
1
Gamma Function
Other Gamma Function Properties:
Start with Weierstrass Gamma Formula

z
zz


sin
1
Euler
Reflection Formula






1
1
k
k
x
x
k
x
e
x
e
x





















1
2
2
2
1
2
1
11
1
kk
k
x
k
x
xx
k
x
x
e
k
x
e
k
x
eex
xx

Use the fact that Γ (-x)=- Γ (1-x)/x to obtain














1
2
2
1
1
1
k k
x
x
xx
Now use the well-known infinite product












1
2
2
1sin
k k
x
xx
q.e.d.

214. 214
SOLO
Primes
Proof





t
t
t
z
td
e
t
z
0
1
Gamma Function
Other Gamma Function Properties:
z
zz


cos2
1
2
1













Start from
Substitute ½ +z instead of z

z
zz


sin
1
z
z
zz




cos
2
1
sin
2
1
2
1



























q.e.d.

215. 215
SOLO
Primes





t
t
t
z
td
e
t
z
0
1
Gamma Function
Duplication and Multiplication Formula:
  0Re2
22
1
12








zzzz
z

Legendre Duplication Formula
1809
Adrien-Marie Legendre

(
1752 – 1833
)
Proof:
   
  2/1,2sin22sin2
2sin22sincos2,
21
2/
0
1221
0
1221
2/
0
1221
2/
0
1212
zBdd
ddzzB
zzzzz
zzzz












 

 
0Re
2/1
2/1
22/1,2,
2
2121







z
z
z
zBzzB
z
zz
zz
We have
therefore
q.e.d

 0Re2
22
1
12
2
1















zzzz
z


216. 216
SOLO





t
t
t
z
td
e
t
z
0
1
Gamma Function
Duplication and Multiplication Formula:
 

znn
n
n
z
n
z
n
zz
znn





 













 2/12/1
2
121

Gauss
Multiplication
Formula
n
z
1

Carl Friedrich Gauss
(1777 – 1855
)


nn
n
nn
n2/1
2121





















 Euler
Multiplication
Formula
Gamma Function

217. 217
SOLO
Primes





t
t
t
z
td
e
t
z
0
1
Gamma Function
Some Special Values of Gamma Function:
q.e.d
 

 








2
222/1
0
2
0
2
2 t
t
u
ut
duudt
t
t
t
udetd
t
e
      


n
n
nnnnn
2
12531
2/12/112/32/12/12/12/1




 
 
 

  



12531
21
2/12/32/1
2/1
2/1
2/3
2/1









nnnn
n
n
nn

2/1
 


n
n
n
2
12531
2/1



 



12531
21
2/1



n
n
nn

Proof:

218. 218
Jacob Bernoulli
1654-1705
The Bernoulli numbers are among the most interesting and important number
sequences in mathematics. They first appeared in the posthumous work "Ars
Conjectandi" (1713) by Jakob Bernoulli (1654-1705) in connection with sums of
powers of consecutive integers. Bernoulli numbers are particularly important in
number theory, especially in connection with Fermat's last theorem (see, e.g.,
Ribenboim (1979)). They also appear in the calculus of finite differences (Nörlund
(1924)), in combinatorics (Comtet (1970, 1974)), and in other fields.
Bernoulli Numbers
The Bernoulli numbers B
n
play an important role in several topics of
mathematics. These numbers can be defined by the power series
SOLO





0 !1
n
n
nz
n
z
B
e
z
Complex Variables

219. 219
SOLO
Bernoulli Numbers





0 !1
n
n
n
seriesTaylor
z
n
z
B
e
z
Let compute the Bernoulli number using
1
Residue2
2
!
12
!
1
1






  z
n
eC
nzn
e
z
i
i
n
z
zd
e
z
i
n
B
z


R

R
C

C
planez
zRe
zIm
The zeros of e
z
= 1 are at z = ± 2 π i k
   





















































 
 
















1 1
1 1
2
1'
22
1'
2
1
2
1
2
1
!
1
lim
1
2
lim
1
lim
1
2
lim2
2
!
1
Residue2
2
!
k k
nn
k k
n
kiz
Hopitall
z
kiz
n
kiz
Hopitall
z
kiz
z
n
e
n
kiki
n
ze
kiz
ze
kiz
i
i
n
e
z
i
i
n
B
z








0
1









x
xn
n
n
e
x
xd
d
B
Complex Variables

220. 220
SOLO
Bernoulli Numbers





0 !1
n
n
nz
n
z
B
e
z
Let compute the Bernoulli number using
1
Residue2
2
!
12
!
1
1






  z
n
eC
nzn
e
z
i
i
n
z
zd
e
z
i
n
B
z


R

R
C

C
planez
zRe
zIm
The zeros of e
z -1
= 1 are at z = ± 2 π i k
   











  





 1 11 2
1
2
1
!
1
Residue2
2
!
k k
nnz
n
e
n
kiki
n
e
z
i
i
n
B
z 


 
 






























oddn
evennk
oddn
evennki
kiki
k
nn
k
n
k
n
k
n
0
12
0
211
1
2/
1
11



















oddn
evennn
n
k
n
B
n
n
k
nn
n
n
0
2
!1
2
1
2
!1
2
2/
1
2/





,2,1,0
120
22
2
!21
2
2









 m
mn
mnm
m
B
m
m
n


Complex Variables




1
1
k
n
k
nwhere is the Zeta Function

221. SOLO
Euler Zeta Function and the Prime History

232
4
1
3
1
2
1
1
In 1650 Mengoli asked if a solution exists for
P. Mengoli
1626 - 1686
The problem was tackled by Wallis, Leibniz, Bernoulli family, without success .
The solution was given by the young Euler in 1735. The problem was named “Basel
Problem” for Basel the town of Bernoulli and Euler.
Euler started from Taylor series expansion of the sine function

!7!5!3
sin
753
xxx
xx
Dividing by x, he obtained

!7!5!3
1
sin
642
xxx
x
x
The roots of the left side are x =±π, ±2π, ±3π,…. However sinx/x is not a
polynomial, but Euler assumed (and check it by numerical computation)
that it can be factorized using its roots as
 



















































2
2
2
2
2
2
9
1
4
11
2
1
2
111
sin

xxxxxxx
x
x
Leonhard Euler
(1707 – 1783
)
Return to Euler
Riemann's Zeta Function

222. SOLO

!7!5!3
1
sin
642
xxx
x
x




























2
2
2
2
2
2
9
1
4
11
sin

xxx
x
x
Leonhard Euler
(1707 – 1783
)
If we formally multiply out this product and collect all the x
2
terms, we
see that the x
2
coefficient of sin(x)/x is










1
22222
11
9
1
4
11
nn

But from the original infinite series expansion of sin(x)/x, the coefficient of x
2
is
−1/(3!) = −1/6. These two coefficients must be equal; thus,




1
22
11
6
1
nn
6
1
2
1
2



nn
Euler extend this to a general function, Euler Zeta Function
  ,4,3,2
4
1
3
1
2
1
1:  nn
nnn

The sum diverges for n ≤ 1 and
converges for n > 1.
Euler computed the sum for n up to n = 26. Some of the values are given here
    ,
9450
8,
945
6,
90
4,
6
2
8642







 
Euler checked the sum
for a finite number of
terms.
EulerZeta Function and the Prime History (continue – 1)
Riemann's Zeta Function

223. SOLO
Euler Product Formula for the Zeta Function
Leonhard Euler proved the Euler product formula for the Riemann
zeta function in his thesis Variae observationes circa series infinitas
(Various Observations about Infinite Series), published by St
Petersburg Academy in 1737
 

 

primep
x
n
x
pn 1
11
1
where the left hand side equals the Euler Zeta Function
Euler Proof of the Product Formula
 
xxxxx
s
8
1
6
1
4
1
2
1
2
1

 






xxxxxxx
x
13
1
11
1
9
1
7
1
5
1
3
1
1
2
1
1 
 






xxxxxxxx
x
33
1
27
1
21
1
15
1
9
1
3
1
2
1
1
3
1

 












xxxxxxx
x
17
1
13
1
11
1
7
1
5
1
1
2
1
1
3
1
1 
all elements having a factor of 3 or 2 (or both) are removed
 


xxxx
n
x
n
x
5
1
4
1
3
1
2
1
1
1
1
 converges for integer x > 1
all elements having a
factor of 2 are
removed
Leonhard Euler
(1707 – 1`783
)
EulerZeta Function and the Prime History (continue – 2)
Riemann's Zeta Function

224. SOLO
Leonhard Euler
(1707 – 1`783
)
Euler Product Formula for the Zeta Function
  

 

primep
x
n
x
pn
x
1
11
1

Euler Proof of the Product Formula (continue)
 












xxxxxxx
x
17
1
13
1
11
1
7
1
5
1
1
2
1
1
3
1
1 
Repeating infinitely, all the non-prime elements are removed, and we get:
1
2
1
1
3
1
1
5
1
1
7
1
1
11
1
1
13
1
1
17
1
1 









































 x
xxxxxxx

Dividing both sides by everything but the ζ(s) we obtain







































xxxxxx
x
13
1
1
11
1
1
7
1
1
5
1
1
3
1
1
2
1
1
1

Therefore
  

 

primep
x
n
x
pn
x
1
11
1

EulerZeta Function and the Prime History (continue – 3)
Riemann's Zeta Function

225. 225
SOLO
Riemann's Zeta Function
The Riemann Zeta Function or Euler–Riemann Zeta Function,
ζ(z), is a function of a complex variable z that analytically
continues the sum of the infinite series
 yixz
n
z
n
z


1
1

“On the Number of Primes Less Than a Given Magnitude”, 7 page
paper offered to the Monatsberichte der Berliner Akademie on
October 19, 1859. The exact publication date is unknown.
  z
z
zz
zz








1
2
sin12
1



To construct the analytic Continuation of the Zeta Function,
Riemann established the relation (see proof ).
where Γ(s) is the Gamma Function, which is an equality of
Meromorphic Functions valid on the whole complex plane. This
equation relates values of the Riemann Zeta Function at the points z
and 1 − z. The functional equation (owing to the properties of sin)
implies that ζ(z) has a simple zero at each even negative integer
z
 = −2n
— these are known as the trivial zeros of ζ(z). For s an even
positive integer, the product sin(πz/2)Γ(1−z) is regular and the
functional equation relates the values of the Riemann Zeta Function
at odd negative integers and even positive integers.
Georg Friedrich Bernhard Riemann
(1826– 1866)

226. SOLO


,2,1
1
1
1




n
n
B
n
nn

B
n are the Bernoulli numbers
Those roots are called the Trivial Zeros
of the Zeta Function. The remaining
zeros of ζ (z) are called Nontrivial Zeros
or Critical Roots of the Zeta Function.
The Nontrivial Zeros are located on a
Critical Strip defined by 0 < x < 1.
Since B
n+1 = 0 for n + 1 odd (n even)
we also have
 ,2,102  mm
  xyixz
pn
z
primep
z
n
s


  


Re
1
11
1

Riemann Zeta Function Zeros
Since the product contains no zero factors
we see that ζ (z) ≠ 0 for Re {z} >1.
Graph showing the Trivial Zeros, the
Critical Strip and the Critical Line
of ζ (z) zeros.
We shall prove that

227. 227
Riemann's Zeta Function

228. 228
Re ζ (z) in the original domain, Re z > 1.
Re ζ (z) after Riemann’s extension.
Riemann's Zeta Function

229. 229
SOLO
The position of the complex zeros can be seen
slightly more easily by plotting the contours of
zero real (red) and imaginary (blue) parts, as
illustrated above. The zeros (indicated as
black dots) occur where the curves intersect
The figures bellow highlight the zeros in
the complex plane by plotting |ζ(z)|) where
the zeros are dips) and 1/|ζ(z)) where the
zeros are peaks).
Riemann's Zeta Function

230. 230
Riemann's Zeta Function
The Riemann Hypothesis
The Non-Trivial Zeros ρ of ζ (z) has Re ρ½ =
This Hypothesis was never proved.

1
z

231. 231
SOLO
 1Re
1
0
1


 



zxfordt
e
t
zz
t
t
t
z






u
u
u
z
du
e
u
z
0
1
Proof:
Gamma Function
Change of variables u=nt 








t
t
nt
z
z
t
t
nt
z
td
e
t
ntdn
e
nt
z
0
1
0
1
Thus for n=1,2,3,…,N

















t
t
Nt
z
z
t
t
t
z
z
t
t
t
z
z
td
e
t
N
z
td
e
t
z
td
e
t
z
0
1
0
2
1
0
1
1
2
1
1
1

0& xyixz
Summing those equations
for x > 0  
















t
t
z
Ntttzzz
tdt
eeeN
z
0
1
2
1111
2
1
1
1
_________________________________________________

Riemann's Zeta Function

232. 232
SOLO
Proof (continue – 1): 0& xyixz
Since converges only for Re (z)= x > 1, then letting N → ∞, we obtain for x > 1



1n
z
n
Uniform convergence of
 

















t
t
z
Nttt
N
zz
tdt
eee
z
0
1
2
111
lim
2
1
1
1


01
1
1
111
1
2
2




tq
eeee
t
q
q
t
q
t
q
t


allows to interchange between limit and the integral :
 RatioGoldentd
e
t
td
e
t
td
e
t
z
t
t
t
zt
t
t
zt
t
t
z
zz














 







2
51
1112
1
1
1
ln2
1ln2
0
1
0
1
























 ln2
0
2
1
ln2
0
1ln2
0
1
11
11
t
t
tt
x
t
t
t
xyixz
t
t
t
z
td
ee
ttd
e
t
td
e
t

The first integral gives
The integral diverges for 0 < x ≤ 1, and converges only for x > 1
 1Re
1
0
1


 



zxfordt
e
t
zz
t
t
t
z

Riemann's Zeta Function

233. 233
SOLO
Proof (continue – 2): 0& xyixz
 



















t
t
t
zt
t
t
zt
t
t
z
zz
td
e
t
td
e
t
td
e
t
z


ln2
1ln2
0
1
0
1
1112
1
1
1

In the second integral we have
This integral converges only for x > 1, therefore we proved that
ln21
2/
 tforee
tt
since RatioGoldeneforee
ttt


 
2
51
01
2/2/
  






















t
t
t
x
termfinite
t
tx
tu
dtedv
t
t
t
xt
t
t
xiyxz
t
t
t
z
td
e
t
xettd
e
t
td
e
t
td
e
t
x
t


 ln2
2/
2
ln2
2/1
ln2
2/
1
ln2
1
ln2
1
122
11
1
2/



  
 
 









finite
t
t
txx
xx
t
t
t
x
td
et
xxxxtermsfinitetd
e
t
 ln2
2/1
ln2
2/
1
1
212
  1Re
12
1
1
1
0
1








 



zxfortd
e
t
zzz
t
t
t
z
zz

 1Re
1
0
1


 



zxfordt
e
t
zz
t
t
t
z

After [x] (the integer defined such that x-[x] < 1) such integration the power of t in
the integrand becomes x-[x]-1 < 0. and we have:
q.e.d.
Riemann's Zeta Function

234. 234
SOLO
Proof
The integral can be rewritten as



00
1sin2
1
0
0
1
itoreturnsandzeroencirclesiatstartspaththe
d
e
i
zz
z
i
i
z




 









i
i
y
x
i
i
2

  

IntegralIII
i
i
z
originaroundCircleIntegralII
i
i
z
IntegralI
i
i
z
i
i
z
d
e
d
e
d
e
d
e




















































1
lim
1
lim
1
lim
1
1
0
1
0
1
0
0
0
1
Riemann's Zeta Function

235. 235
SOLO
Proof (continue – 1)
The first integral can be written as
i
i
y
x
i
i
2
  





















 








 t
t
t
z
zi
e
t
t
t
z
zi
t
t
it
z
i
it
i
i
z
td
e
t
etd
e
t
etd
e
eit
d
e
i
0
1
1
0 1
1
1
0
0 1
0 111
lim
1
lim














The second integral can be written as
  
  
0
1
2
lim2
1
2
lim
2
1
2
lim
1
lim
2
0
20
2
0
2
1
0
2
0
2
1
0
2
1
0




























































d
e
dei
e
e
dei
e
e
d
e
ii
i
i
e
x
i
e
iyx
i
i
e
iyx
ie
originaroundCircle
i
i
z




00
1sin2
1
0
0
1
itoreturnsandzeroencirclesiatstartspaththe
d
e
i
zz
z
i
i
z




 









Riemann's Zeta Function

236. 236
SOLO
Proof (continue – 2)
The third integral can be written as
i
i
y
x
i
i
2
  



















 








 t
t
t
z
zi
e
t
t
t
z
zi
t
t
it
z
i
it
i
i
z
td
e
t
etd
e
t
etd
e
eit
d
e
i
0
1
1
0
1
1
1
0
1
0 111
lim
1
lim














Therefore

 
 





















t
t
t
zt
t
t
zzizit
t
t
z
zizi
i
i
z
td
e
t
zitd
e
t
i
ee
itd
e
t
eed
e
0
1
0
1
0
10
0
1
1
sin2
12
2
11







But we found that
 1Re
1
0
1


 



zxfordt
e
t
zz
t
t
t
z




00
1sin2
1
0
0
1
itoreturnsandzeroencirclesiatstartspaththe
d
e
i
zz
z
i
i
z




 









Therefore 










0
0
1
1sin2
1
i
i
z
d
e
i
zz
z







The right hand is analytic for any z ≠ 1. Since it equals Zeta Function in the half
plane x > 1, it is the Analytic Continuation of Zeta to the complax plane for any z ≠ 1 .











0
0
1
1sin2
1
i
i
z
d
e
i
zz
z







q.e.d.
Riemann's Zeta Function

237. 237
SOLO
Proof

 












0
0
1
1
2
sin22
1
i
i
z
z
z
z
id
e






 R

RC
C
plane
Re
Im
Let add a circular path of radius R → ∞. On this path
  
0
1
lim
1
2
0
1










  







d
e
eR
d
e
i
i
i
R
eR
z
i
R
eR
deRd
C
z
Therefore we have
Since the integral is over a closed path in the complex λ plane, we can use the
Residue Theorem to calculate it. The residues are given by
,2,121  nnie 

 
   















1
1
1
1
10
0
1
2222
11
n
z
n
z
zi
i
z
niiniid
e
d
e





   


























d
e
d
e
d
e
d
e
z
C
zi
i
zi
i
z
R
1111
110
0
10
0
1
Riemann's Zeta Function

238. 238
SOLO
Proof (continue)

 












0
0
1
1
2
sin22
1
i
i
z
z
z
z
id
e







R

R
C
C
plane
Re
Im

    
















1
1
11
1
1
1
1
0
0
1
1
22222
1
n
z
zzz
n
z
n
z
i
i
z
n
iiiniiniid
e



 

 
  
  







2
sin2
2/2/lnln11 z
eeieeiiiiiii
izizizizzzzz 

z
n
n
z




1
1
1
1


 z
z
id
e
z
i
i
z













1
2
sin22
1
0
0
1





q.e.d.
Riemann's Zeta Function

239. 239
SOLO
i
i
y
x
i
i
2

 
00
12
1
0
0
1
itoreturnsandzeroencirclesiatstartspaththe
d
ei
z
z
i
i
z



 









We also found

 z
z
id
e
z
i
i
z













1
2
sin22
1
0
0
1





Has zeros for
,...4,2,00
2
sin 





zfor
z
 ,7,5,301  zforz
z1 Has no zeros, but has simple poles for z = 1,2,3,4.…,
If we return to ζ (z) equation we can see that the zeros of
are cancelled by the poles of Γ (1-z). Only the simple pole
at z = 1 remain and is the single pole of ζ (s) .







0
0
1
1
i
i
z
d
e



Let find the Residue of this pole:
 





 


0
0
1
111 12
1
lim11lim1lim
i
i
z
zzz
d
ei
zzzz









 



1
cos
lim
sin
1
lim11lim
1
'
1
1
sin
1
1






 zzz
z
zz
z
HopitalL
z
z
zz
z 














0
0
1
1 12
1
lim
i
i
z
z
d
ei






Riemann's Zeta Function

240. 240
SOLO
Proof
 z
z
zzz
z






 1
2
sin22sin2 



 z
z
id
e
z
i
i
z













1
2
sin22
1
0
0
1





We found











0
0
1
1sin2
1
i
i
z
d
e
i
zz
z







Combining those two relations, we get
 z
z
zzz
z






 1
2
sin22sin2 


q.e.d.
Riemann's Zeta Function

241. 241
SOLO
Proof
  zzzz
zz


112/sin2
1

Start from
use
 z
z
zzz
z






 1
2
sin22sin2 



z
zz


sin
1 
z
zz


1
sin


or

 z
z
z
z
z








1
2
sin2
1




  z
z
zz
zz








1
2
sin12
1



q.e.d.
Return to Riemann Zeta Function
Riemann's Zeta Function

242. 242
SOLO
Proof
Start from
use
 z
z
zzz
z






 1
2
sin22sin2 



z
zz


sin
1

z
zz


1
sin





















2
1
2
2
sin
zz
z 

z
z

2


 
z
z
zz
z
z
z









1
2
sin
2/12/
1
2
1
1



or



 
z
z
zzz
zzz




1
2/1
1
122/
2/12/12/




 


z
z
z
z
zzzz



1
2/12/
12/12/


Riemann's Zeta Function

243. 243
SOLO
Proof (continue)



 


z
z
z
z
zzzz



1
2/12/
12/12/


or



 
z
z
zzz
zzz




1
2/1
1
122/
2/12/12/

 













2
1
2
2
12/1 zz
z
z

  2/1
2
1
21
2/1
z
z
z
z








 z
z


1

 









2
1
2/1
1
12
2/1 z
z
z
z

therefore
q.e.d.


z
z
zz
zz








1
2
1
2/
2/12/

Use Legendre
Duplication Formula:   0Re2
22
1
12








zzzz
z

2/z
z

Riemann's Zeta Function

244. 244
SOLO
Proof
i
i
y
x
i
i
2



00
1sin2
1
0
0
1
itoreturnsandzeroencirclesiatstartspaththe
d
e
i
zz
z
i
i
z




 









We found
and 
zz
z




sin
1
z
zz


sin
1

 
0
0
12
1
0
0
1
itoreturnsand
zeroencirclesiatstartspaththe
d
ei
z
z
i
i
z




 









 











0 1
21
12
!1
12
!1
i
i
n
C
nzn d
ei
n
z
zd
e
z
i
n
B 



i
i
y
x
i
i
2
therefore


1
!1
1
1




n
n
B
n
z
z

 
0
0
12
1
0
0
1
itoreturnsand
zeroencirclesiatstartspaththe
d
ei
z
z
i
i
z




 









zz 
nz

1
1
1



n
B
n
nn



,2,1,0
1
1
1




n
n
B
n
nn
 B
n
are the Bernoulli numbers
q.e.d.
We found
Zeta-Function Values and the Bernoulli Numbers
Return to Riemann Zeta Function Zeros
Riemann's Zeta Function

245. 245
SOLO
Zeta Function Values and the Bernoulli Numbers



,3,2,1
!22
21
2
2
2


 mB
m
m
m
mm




z
z
zz
zz








1
2
1
2/
2/12/
Let use
with z = 2 m 
 
 m
m
mm
mm
21
2
21
2
2/21









 

 
 


 


 
 
m
mm
m
mmm
m
m
B
mm
m
B
mm
m
m
m
m
m
2
2/112
2
22/12
2/1
2/1!2
!121
!22
21
2/1
!1
2
2/1
!1
21


















We found
 
 
 
,3,2,1
2/1!2
!121
21
2
2/112





mB
mm
m
m
m
mm


Riemann's Zeta Function

246. 246
SOLO
Zeta Function Values and the Bernoulli Numbers
We found
 

 
  
    
 
 
2/1
12
2/1
1
2/1
!12
!121
224212531
121221
12531
21
2/1















m
m
mm
m
m
m
mm
mmmmm





 
 
 
,3,2,1
2/1!2
!121
21
2
2/112





mB
mm
m
m
m
mm

Therefore
Finally


,2,1,0
1
1
1




n
n
B
n
nn

  ,3,2,1
2
1
21
2
 mB
m
m
m

We also found
The two expressions
Agree .
Riemann Zeta Function
Riemann's Zeta Function

247. 247
SOLO
Hadamard Infinite Product Expansion of Zeta Function
Graph showing the Trivial Zeros, the
Critical Strip and the Critical Line
of ζ (z) zeros.
 


















1
1
1
2
1
1
1
2
2
1
11
10
j
a
z
pa
z
a
z
j
zczc
p
j
p
jj
p
p
e
a
z
efzf


Since (z-1) ζ (z) is Analytic and has only Zeros we can use the Hadamard Infinite
Product Expansion
Zeta Function ζ (z) has Order p=0, and

2
1
2
1
0
1
B
The Zero of the Zeta Function ζ (z) are
-Trivial Zeros at z = -2n, n=1,2,…
- Nontrivial Zeros ρ on the Critical Zone 0 < Re ρ < 1






























1
2
0
10
2/10
2
1101
1
n
n
z
zofzeros
trivial
z
zofzeros
nontrivial
zc
fzf
e
n
z
e
z
ezz









Hadamard Infinite Product Expansion of (z-1) ζ (z) is:


pi
i
f
f
zd
d
c
z
i
i
i ,,1,0
!1
:
0
1
1 












 


12ln
2/1
2/2ln2/1
0
0'0
:
1
0
1
1 




















zzzf
z
f
f
c 


2
1
0&
2
2ln
0'  


Riemann's Zeta Function

248. 248
SOLO
Hadamard Infinite Product Expansion of Zeta Function (continue)
Graph showing the Trivial Zeros, the
Critical Strip and the Critical Line
of ζ (z) zeros.

 


























1
2
0
10
12ln
2
11
2
1
n
n
z
zofzeros
trivial
z
zofzeros
nontrivial
z
e
n
z
e
ze
zz
























1
22
2
1
22/
1
n
n
z
z
e
n
z
e
z
z

Hadamard Infinite Product Expansion of (z-1) ζ (z) is:
We found the Weierstrass Expansion for the Gamma Function:

 
 
















10
0
2/12ln
1
2/112






Re
z
z
e
z
zz
e
z






























2
1
22
2
1
22
1
2
z
e
zz
e
e
n
z
zz
n
n
z

Hadamard (1893) used the Weierstrass product theorem to
derive this result. The plot above shows the convergence of the
formula along the real axis using the first 100 (red), 500
(yellow), 1000 (green), and 2000 (blue) Riemann zeta function
zeros.
Riemann's Zeta Function

249. Fourier Transform
   


 dttjtftfF  exp:F
SOLO
Jean Baptiste Joseph
Fourier
1768 - 1830
F (ω) is known as Fourier Integral or Fourier Transform
and is in general complex
     jAFjFF expImRe 
Using the identities
 t
d
tj 


 

 2
exp
we can find the Inverse Fourier Transform Ftf
-1
F
   
     00
2
1
2
exp
2
expexp
2
exp






















tftfdtfd
d
tjf
d
tjdjf
d
tjF










  






2
exp:
d
tjFFtf
-1
F
  00
2
1



tftfdtf 
If f (t) is continuous at t, i.e. f (t-0) = f (t+0)
This is true if (sufficient not necessary)
f (t) and f ’ (t) are piecewise continue in every finite interval1
2 and converge, i.e. f (t) is absolute integrable in (-∞,∞)


dttf

250. Fourier Transform
SOLO
tf
-1
F
F
FProperties of Fourier Transform
Linearity 1
        
221122112211 exp: FFdttjtftftftf 



F
Symmetry 2
tF
-1
F
F
f2
       tFdttjtFf
dt
tjtFf
d
tjFtf
t
F















exp2
2
exp
2
exp
Proof:
Conjugate Functions3
tf
*
-1
F
F

*
F
Proof:
    tf
d
tjF
d
tjFtf
****
2
exp
2
exp
1-
F














251. Fourier Transform
     


















a
F
aa
d
a
jfdttjtaftaf
ta





1
expexp:F
         

 FjdttjjtfF
d
d
dttjtftfF
nn
n
n






FF expexp:
SOLO
tf
-1
F
F
FProperties of Fourier Transform
Scaling4
Derivatives5
Proof:
taf
-1
F
F






a
F
a
1
Proof:
Corollary: for a = -1
tf
-1
F
F
F
tftj
n

-1
F
F


F
d
d
n
n
tf
td
d
n
n
-1
F
F
Fj
n
      





 Fj
d
tjjFtf
td
dd
tjFFtf
nn
n
n
1-1-
FF 





2
exp
2
exp

252. Fourier TransformSOLO
tf
-1
F
F
FProperties of Fourier Transform
Convolution6
Proof:
       
      


212121
212121
expexpexp
expexpexp:
FFFdjfdduujufjf
ddttjtfjfdtdtfftjdtff
ut








































 
  

















F
tftf
21
-1
F
F

21
*FF 



  dtfftftf
2121
:*
-1
F
F

21
FF
The animations above graphically illustrate the convolution of two rectangle functions (left) and two
Gaussians (right). In the plots, the green curve shows the convolution of the blue and red curves as a
function of t, the position indicated by the vertical green line.
The gray region indicates the product as a function of g (τ) f (t-τ) , so its area as a function of t is
precisely the convolution.
http://mathworld.wolfram.com/Convolution.html

253. Fourier Transform
SOLO
tf
-1
F
F
FProperties of Fourier Transform
 





 

dFFdttftf 2
*
1
2
*
1
2
1
Parseval’s Formula7
Proof:
  



 dttjtfF  exp
11
      























22
exp
2
exp 2
*
11
2
*
2
*
1
2
*
1
d
FF
d
dttjtfFdt
d
tjFtfdttftf
   























22
exp
2
exp
21122121
d
FF
d
dttjtfFdt
d
tjFtfdttftf
  







2
exp
*
2
*
2
d
tjFtf
    





 dttjtfFdttjtfF  expexp
1111
  







 



dFFdFFdttftf
212121
2
1
2
1

254. Signal Duration and Bandwidth
SOLO
tf
-1
F
F
FRelationships from Parseval’s Formula
 




 

dFFdttftf 2
*
1
2
*
1
2
1
Parseval’s Formula7
Choose tstjtftf
m

21


,2,1,0
2
1
2
2
2





nd
d
Sd
dttst
m
m
m




tftj
n

-1
F
F


F
d
d
n
n
and use5a
Choose 

n
n
td
tsd
tftf 
21 and use5b
tf
td
d
n
n
-1
F
F
Fj
n

 ,2,1,0
2
1 2
2
2





ndSdt
td
tsd
m
n
n


Choosec
 



 ,2,1,0,,2,1,0
2
* 











mnd
d
Sd
S
j
dt
td
tsd
tstj
m
m
n
n
n
n
mm







n
n
td
tsd
tf
1
tstjtf
m

2

255. Fourier Transform
SOLO
tf
-1
F
F
FProperties of Fourier Transform
Modulation9
Shifting: for any a real 8
Proof:
 ttf
0cos
-1
F
F
   
00
2
1
  FF
Proof:
   tjtjt
000
expexp
2
1
cos  
atf
-1
F
F
  ajF exp tajtfexp
-1
F
F
aF
        

Fajdajfdttjatfatf
at







expexpexp:F
       aFdttajtfdttjtajtftajtf 





 expexpexp:expF
use shifting property with a=±ω
0

256. atf
-1
F
F
  ajF exp
Fourier Transform
SOLO
tf
-1
F
F
FProperties of Fourier Transform (Summary)
Linearity 1
        
221122112211 exp: FFdttjtftftftf 



F
Symmetry 2
tF
-1
F
F
f2
Conjugate Functions3 tf
*
-1
F
F

*
F
Scaling4 taf
-1
F
F






a
F
a
1
Derivatives5
tftj
n

-1
F
F


F
d
d
n
n
tf
td
d
n
n
-1
F
F
Fj
n
Convolution6
tftf
21
-1
F
F

21
*FF 



  dtfftftf
2121 :*
-1
F
F

21
FF
 




 

dFFdttftf 2
*
1
2
*
1
2
1
Parseval’s Formula7
Shifting: for any a real 8
tajtfexp
-1
F
F
aF
Modulation9
 ttf
0cos
-1
F
F
   
00
2
1
  FF
  







 



dFFdFFdttftf
212121
2
1
2
1

257. Laplace’s Transform
C2



f
a
0t
0
0


t
js
s - plane
SOLO
Laplace L-Transform (continue – 1)
The Inverse Laplace’s Transform (L
-1
) is given by: 



j
j
ts
dsesF
j
tf
2
1
Using Jordan’s Lemma (see “Complex Variables” presentation or the end of this one)
Jordan’s Lemma Generalization
If |F (z)| ≤ M/R
k
for z = R e
iθ
where k > 0 and M are constants, then
for Γ a semicircle arc of radius R, and center at origin:
 00lim 



mzdzFe
zm
R
where Γ is the semicircle, in the left part of z plane.
x
y

R
we can write
   





j
j
tsts
f
f
dsesF
j
dsesF
j
sFtf


 2
1
2
11-
L
    



dsesF
j
dsesF
j
dsesF
j
sFtf
ts
C
ts
j
j
ts
 2
1
2
1
2
1
0

1-
L
If the F (s) has no poles for σ > σ
f+
, according to Cauchy’s Theorem
we can use a closed infinite region to the left of σ
f+
, to obtain

258. Laplace’s Transform
SOLO
Properties of Laplace L-Transform
s - Domain
t - Domain
tf   




f
st
sdtetfsF Re
0
1 
if
M
i
ii
zsFc maxRe
1


Linearity 

M
i
iitfc
1
3  





 000
1121 nnnn
ffsfssFs 
Differentiation

n
n
td
tfd
4






t
t
df
ss
sF

0
lim
1
Integration 

t
df
5

s
sF
Real Definite
Integration

t
df
0


t
ddf
00

 
2
s
sF
2 





a
s
F
a
1
Scaling taf

259. Laplace’s Transform
SOLO
Properties of Laplace L-Transform (continue – 1)
s - Domain
t - Domain
tf   




f
st
sdtetfsF Re
0
6

n
n
sd
sFd
Muliplicity by t
n
tft
n

7 

0
dssF
Division by t

t
tf
8 sFe
sTime shifting tutf
9 asF
Complex
Translations
tfe
ta
10
sHsF
Convolution
t - plane
 


0
 dthfthtf
11  



j
j
dsHF
j
sHsF
j



 2
1
2
1
Convolution
s - plane
thtf

260. Laplace’s Transform
SOLO
Properties of Laplace L-Transform (continue – 2)
s - Domain
t - Domain
tf   




f
st
sdtetfsF Re
0
12Initial Value Theorem  sFstf
st 


limlim
0
13Final Value Theorem  sFstf
st 0
limlim


14Parseval’s Theorem
  
 






 




j
j
j
j
ts
j
j
ts
dssGsF
j
dsdtetgsF
j
dttgdsesF
j
dttgtf








2
1
2
1
2
1
0
00

261. SOLO
Z- Transform and Discrete Functions
Z Transform
The Z- Transform (one-sided) of a sequence { f (nT); n=0,1,… } is defined as :
  




0
:
n
n
zTnfzFTnfZ
where T, the sampling time, is a positive number.
tf
  



0n
T Tntt
  



0
*
n
T TntTnfttftf 
tf
*
tf
T t

262. tf
  



0n
T Tntt
  



0
*
n
T
TntTnfttftf 
tf
*
tf
T t
   



f
ts
dtetftfsF
0
L
SOLO
Sampling and z-Transform
     
















0
1
1
00
sT
n
sTn
n
T
e
eTnttsS LL
 
  
  






























0
00
**
1
1
2
1







 f
j
j
tsT
n
sTn
n
d
e
F
j
ttf
eTnfTntTnf
tfsF
L
L
L














































ts
e
ofPoles
tsts
F
ofPoles
tsts
n
n
sT
e
F
Resd
e
F
j
e
F
Resd
e
F
j
eTnf
sF












1
1
0
*
112
1
112
1
2
1
Poles of
Ts
e

1
1
Poles of
F
planes
T
ns
n


2

j
j
0s
Laplace Transforms
The signal f (t) is sampled at a time period T.
1

2

R
R
Poles of
Ts
e

1
1
Poles of
F
plane
T
ns
n


2

j
j
0s
Z Transform

263. tf
  



0n
T Tntt
  



0
*
n
T
TntTnfttftf 
tf
*
tf
T t
SOLO
Sampling and z-Transform (continue – 1)





 
























































nn
Ts
e
n
ts
T
n
js
T
n
js
e
ofPoles
ts
T
n
jsF
TeT
T
n
jsF
T
n
jsF
e
T
n
js
e
F
RessF
ts
n
ts














21
2
lim
2
1
2
lim
1
1
2
2
1
1
*
Poles of
F
j

0s
T
2
T
2
T
2
Poles of

*
F
plane
js 
The signal f (t) is sampled at a time period T.
The poles of are given by
ts
e

1
1


T
n
jsnjTsee
n
njTs 

 2
21
2












n T
n
jsF
T
sF
21
*
Z Transform

264. SOLO
FF
-1
frequency
-B/2 B/2
B
FF
-1
-B/2 B/2
B
1/T
s-1/T
s frequency
Sample
Sampling a function at an interval T
s
(in time domain)
Anti-aliasing filters is used to enforce band-limited assumption.
causes it to be replicated
at
1/ T
s
intervals in the other (frequency) domain.
Sampling and z-Transform (continue – 2)
Bandlimited Continuous Time Signal
1/B sec
a
m
p
l
i
t
u
d
e
time (sec)
-0.4
-0.2
0.2
0
0.4
0.6
0.8
1
0 510 15-15-10-5
Discrete-Time (Sampled) Signal
a
m
p
l
i
t
u
d
e
sample
-0.4
-0.2
0.2
0
0.4
0.6
0.8
1
0
10
20
-20 -10
Z Transform

265. tf
  



0n
T Tntt
  



0
*
n
T
TntTnfttftf 
tf
*
tf
T t
SOLO
Sampling and z-Transform (continue – 3)
0z
planez
Poles of
zF
C
The signal f (t) is sampled at a time period T.
The z-Transform is defined as:
  

 




















i
F
iF
i
i
F
Ts
FofPoles
T
F
n
n
ze
ze
F
zTnf
zFsFtf





1
0
*
1
lim:Z












00
0
2
1
1
n
RzndzzzF
jTnf
fC
C
n

Z Transform

266. SOLO
Sampling and z-Transform (continue – 4)
 












0
* 21
n
n
sT
n
eTnf
T
n
jsF
T
sF

We found
The δ (t) function we have:
1



dtt   fdtttf 



The following series is a periodic function:  

n
Tnttd :
therefore it can be developed in a Fourier series:
  







n
n
n T
tn
jCTnttd  2exp:
where:

T
dt
T
tn
jt
T
C
T
T
n
1
2exp
1
2/
2/











Therefore we obtain the following identity:
 







nn
TntT
T
tn
j 2exp
Second Way
Z Transform

267.    



 dttjtftfF  2exp:2 F
 












0
* 21
n
n
sT
n
eTnf
T
n
jsF
T
sF

    



  dtjFFtf 2exp2:2
-1
F
SOLO
Sampling and z-Transform (continue – 5)
We found
Using the definition of the Fourier Transform and it’s inverse:
we obtain  



  dTnjFTnf 2exp2
      







0
111
0
*
exp2exp2exp
nn
n
sTndTnjFsTTnfsF 
     
 





111
*
2exp22  dTnjFjsF
n
     

























nn T
n
F
T
d
T
n
T
FjsF  2
11
22
111
*
We recovered (with –n instead of n)









n T
n
jsF
T
sF
21
*
Second Way (continue)
Making use of the identity: with 1/T instead of T
and ν - ν
1
instead of t we obtain:
 







nn T
n
T
Tnj
11
1
2exp 
 







nn
TntT
T
tn
j 2exp
Z Transform

268. Z Transform
SOLO
Properties of Z-Transform Functions
Z - Domain
k - Domain
kf  




 ff
k
k
rzrzkfzF
0
1 



ii ff
M
i
ii
rzrzFc minmax
1
Linearity 

M
i
iikfc
1
2  ,2,10 kkfmkf zFz
m
Shifting
mkf  





m
k
kmm
zkfzFz
1
mkf  





m
k
kmm
zkfzFz
1
1kf 0fzFz 
3 Scaling kfa
k  





 ff
k
k
razrazakfzaF
0
11

269. Z Transform
SOLO
Properties of Z-Transform Functions (continue – 1)
4Periodic Sequence
kf




111
1
ffN
N
rzrzF
z
z
N = number of units in a period
R
f1-

,+
= radiuses of convergence in F
(1)
(z)
F
(1) (z) = Z -Transform of the first period
5Multiplication by k kfk



ff
rzr
zd
zFd
z
6Convolution 



0
:
m
mkhmfkhkf     


hfhf
rrzrrzHzF ,min,max
7Initial Value  zFf
z
lim0
8Final Value   existsfifzFzkf
zk


1limlim
1
Z - Domain
k - Domain
kf  




 ff
k
k
rzrzkfzF
0

270. Z Transform
SOLO
Properties of Z-Transform Functions (continue – 2)
9Complex Conjugate kf
*



ff
rzrzF
**
10Product khkf 


 hfhf
C
rrzrr
z
zd
zHzF
j
,1,
2
1
1

12Correlation
   1,1,
2
1
11
0





 krrzrr
z
zd
zzHzF
j
kmhmfkhkf
hfhf
C
k
m 
11Parceval’s Theorem
 




  hfhf
Ck
rrzrr
z
zd
zHzF
j
khkf ,1,
2
1
1
0 
Z - Domain
k - Domain
kf  




 ff
k
k
rzrzkfzF
0

271. Z Transform
SOLO
Table of Z-Transform Functions
Z - Domain
k - Domain
kf  
f
k
k
RzzkfzF 



0
1
mkf    110
11


mfzfzfzFz
mm
2
mkf zFz
m
3
kfkfkf  1:  01 fzzFz 4
kfkfkfkf  122:
2
 1021
2
fzfzzzFz 5
kf
3
   2130331
23
fzfzzfzzzzFz 6

272. 272
SOLO
Mellin Transform
 



0
1
xdxfxsFxf
s
MM
We can get the Mellin Transform from the two side Laplace Transform
Robert Hjalmar Mellin
( 1854 – 1933)
  



 xdxfesFxf
sx
2
LL
2
  

 1
0
11
0
1
 




sFxdxfxxdxfxxxfx
ss
MM
 




ic
ic
s
sdsFx
i
x
M
1-
fsfM
2
1
Example:
 sxdexe
xsx



0
1
M

x
exf



273. 273
SOLO
Mellin Transform (continue – 1)
 



0
1
xdxfxsFxf
s
MM
Relation to Two-Sided Laplace Transformation
Robert Hjalmar Mellin
( 1854 – 1933)
tdexdex
tt 
,
Let perform the coordinate transformation


  








 tdeeftdeeftdeefesF
tsttstttst
0
1
M
After the change of functions 
t
eftg

:
   






 tdetgsGtdeefsF
tstst
2LM
Inversion Formula
    xfefsdxsF
i
sdesG
i
tg
xe
t
ic
ic
s
ex
ic
ic
ts
tt









  ML
L
2
1
2
 2
1
2
1
c

t
 cxdssFx
R
C
s


0
M
R
C
R
Mellin Transform

274. 274
SOLO
Properties of Mellin Transform (continue – 2)
  
 
 
 
 
 
 
 
  
f
kk
k
k
fk
k
k
k
f
z
fk
k
k
f
a
f
f
s
SszsFstf
td
d
t
sksksks
SkszsFkstf
td
d
SzszsFCztft
SssF
sd
d
tft
SsasFaRatf
SsFaataf
SsFtf
HolomorphyofStriptdtftsFtftf





















M
M
M
M
M
M
M
M
M0t,
1
11:
1
,
ln
0,,
0,
11
1
0
1

Original Function Mellin Transform Strip of Convergence
Mellin Transform

275. 275
SOLO
Properties of Mellin Transform (continue – 3)
  
 
  
 
  
 
 
 
  
21
0
21
1
0
1
0
1
//
1
1
11:
1
11:
1
ff
t
t
k
f
kk
k
k
k
k
fk
kk
k
k
f
s
SSssFsFxxdxtfxf
sFsxdxf
sFsxdxf
kssss
SssFstf
td
d
t
sksksks
SssFkstft
td
d
SsFtf
HolomorphyofStriptdtftsFtftf





















M2M1
M
M
M
M
M
M
M0t,
Original Function Mellin Transform Strip of Convergence
Mellin Transform

276. 276
Hilbert Transform
SOLO
F (u) a analytical function in the right half u plan including infinity.
According to Cauchy theorem:





C
dz
sz
zF
j
sF
2
1
s
R

1
C
2
C
s
*
s
*
s
complex
plane
complex
plane
Let take the point –s* , where s* is the complex conjugate of s. Since –s* is outside the
contour C, we have




C
dz
sz
zF
j *2
1
0

By adding and subtracting those two relations we obtain:
  




















CC
dzzF
szszj
dzzF
szszj
sF
*
11
2
1
*
11
2
1

where C = C
1
+ C
2
is a closed curve composed by
- C
1
a semicircle in the right half plane
- C
2
a straight line on the imaginary axis of the
complex plane
Augustin Louis Cauchy
(

1789-1857)

277. 277
SOLO
Let compute the integrals:
s
R

1
C
2
C
s
*
s
*
s
complex
plane
complex
plane
along C
1
, assuming R → ∞ we have
jRszsz exp
1
*
11




  




















CC
dzzF
szszj
dzzF
szszj
sF
*
11
2
1
*
11
2
1

djRjzd exp
    0exp2
2
1
lim
*
11
2
1
2
2
1
1














atanalyticF
R
C
FdjRFj
j
dzzF
szszj
I




0
*
11
2
1
1
3 










C
dzzF
szszj
I

along C
2
, assuming R → ∞ we have

 



























j
j
vjz
js
js
j
jC
vdvjF
v
vj
zdzF
szsz
ssz
j
dzzF
szszj
I
22
*
2
1
*
*2
2
1
*
11
2
1
2






 

























j
j
vjz
js
js
j
jC
vdvjF
v
dzzF
szsz
ss
j
dzzF
szszj
I
22
*
4
1
*
*
2
1
*
11
2
1
2





















j
j
j
j
vdvjF
v
vdvjF
v
vj
sF
2222
11





Hilbert Transform

278. 278
SOLO
Let write



 












j
j
j
j
vdvjF
v
FvdvjF
v
vj
sF
2222
11





        jFjjFjF  ImRe
     


    

    













j
j
j
j
vdjFjjF
v
vdjFjjF
v
vj
jFjjF









ImRe
1
ImRe
1
ImRe
22
22
We obtain
By equaling the real and imaginary parts we obtain
  


 

 










j
j
j
j
vdvjF
v
vdvjF
v
v
jF Re
1
Im
1
Re
2222






  


 

 










j
j
j
j
vdvjF
v
vdvjF
v
v
jF Im
1
Re
1
Im
2222






From those relation we can see that if F (s) is analytic in the right half plane,
then it is enough to know it’s value on the imaginary axis to compute F (s) in
the entire right half plane.
Hilbert Transform

279. 279
SOLO
We are interested in cases when σ = 0, i.e. points on the imaginary axis. In this
case:
  


 

 










j
j
j
j
vdvjF
v
vdvjF
v
v
jF Re
1
Im
1
Re
2222






  


 

 










j
j
j
j
vdvjF
v
vdvjF
v
v
jF Im
1
Re
1
Im
2222






It seams that we have a singular point ν = ω, on the path of integration, but we
will see how this can be taken care.
 
 





j
j
vd
v
vjF
jF


Im1
Re
 
 





j
j
vd
v
vjF
jF


Re1
Im
Hilbert Transform

280. 280
SOLO
Suppose that F (z) is an analytic function
on the lower (or upper) half complex plane .
Re
Im


R
R
R
'C
C

We can write
    














R
CRC
d
jF
d
jF
d
jF
d
jF
d
jF

















'
0

0lim
'


C
d
jF




Now

 

 












jFjdjFj
e
ed
jF
d
jFd
jF
C
j
j
CC



 
2
Therefore 
 

















R
R
R
d
jF
d
jFj
jF











0
limlim
Define Cauchy Principal Value  















R
R
R
R
dqdqdqPV




0
lim:



















 d
jF
PV
j
jF
From the development we can see that the limit exist and are
finite, therefore we removed the singularity at ν = ω.
Augustin Louis Cauchy
(

1789-1857)
Hilbert Transform

281. 281
SOLO
Suppose that F (z) is an analytic function
on the lower (or upper) half complex plane .
We can write
    
   





























 d
vjF
PV
j
d
vjF
PVjFjjFjF
ReIm1
ImRe
Comparing real and imaginary parts we obtain
 
 
  
 
 
  





jFd
vjF
PVjF
jFd
vjF
PVjF
Re
Re1
Im
Im
Im1
Re
H
H




























Where H stands for Hilbert Transform.



















 d
jF
PV
j
jF
Re
Im


R
R
R
'C
C

David Hilbert
1862 - 1943
Return to Table of Contents
Hilbert Transform

282. 282
SOLO
References

[2] Churchill, R.V., “Complex Variables and Applications”, McGraw-Hill, Kõgakusha’
1960
[3] Spiegel, M.R., “Complex Variables with an introduction to Conformal Mapping
and its applications”, Schaum’s Outline Series, McGraw-Hill, 1964
[4] Hauser, A.A., “Complex Variables with Physical Applications”, Simon & Schuster,
1971
[5] Fisher, S.D., “Complex Variables”, Wadsworth & Brooks/Cole Mathematics Series,
1986
Complex Variables
[6] Tristan, N., “Visual Complex Analysis”, Clarendon Press, Oxford, 1997
[1] E.C. Titchmarsh, “The Theory of Functions”, Oxford University Press, 2
nd
Ed., 1939
http://www.ima.umn.edu/~arnold/complex.html

283. SOLO
References (continue - 1)

Complex Variables
F.B. Hildebrand, “Advanced Calculus for Applications”, 2
nd
Ed., Prentice Hall,
1976, Ch.10, “Functions of a Complex Variable”
http://en.wikipedia.org/wiki/
G.B. Arfken, H.J. Weber, “Mathematical Methods for Physicists”, Academic Press,
Fifth Ed., 2001
Sokolnikoff, I.S., Redheffer, R.M., “Mathematics of Physics and Modern Engineering”,
2
nd
Ed., McGraw Hill, Kõgakusha, 1966
Walter Rudin, “Real and Complex Analysis”, 2
nd
Ed., McGraw Hill,1974
Marco M. Peloso, “Complex Analysis”, January 21, 2011, University of Milano
http://www.math.umn.edu/~garrett/m/complex/
Wilhelm Schlag, “A Concise Course in Complex Analysis and Riemann Surfaces”,
University of Chicago
Alexander D. Poularikas, Ed., “Transforms and Applications Handbook”, 3th
Edition, CRC Press, 2010

284. SOLO
References (continue -2)

Complex Variables
S. Hermelin, “Fourier Transform”
S. Hermelin, “Gamma Function”
S. Hermelin, “Primes”
S. Hermelin, “Hilbert Transform”
S. Hermelin, “Z Transform”

285. October 1, 2024 285
SOLO
Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 – 2013
Stanford University
1983 – 1986 PhD AA
Complex Variables

286. 286
SOLO
Laplace Fields (general three dimensional)
Vector Analysis
A vector field is said to be a Laplace Field if rAA


0rA

In this case we have
and
 0
22
00
2








AAAAA
0 rA

Harmonic Functions
A continuous function φ with continuous first and second partial derivatives is said
to be harmonic if it satisfies Laplace’s Equation 0
2

Properties of Harmonic Functions
Pierre-Simon Laplace
1749-1827
0
22
 
2
1
0



S
dS
n


n
i
i
SS
1

i
S
n
S
dV
dSn

1
V
F
r

S
r

F
0r
SF
rrr








SS
dS
n
dS
n




General Three Dimensional Complex function
If φ ‘(z) is analytical inside and on C

0

C
dz
zd
zd
Cauchy’s Th.
C
R
If φ ,φ’, ψ, ψ’ are analytical inside and on C



CCC
dz
zd
d
dz
zd
d
dz
zd
d 




0
see
Vector Analysis.ppt

287. 287
SOLO Vector Analysis
Harmonic Functions (continue 1)
A continuous function φ with continuous first and second partial derivatives is said
to be harmonic if it satisfies Laplace’s Equation 0
2

Properties of Harmonic Functions (continue – 1)

n
i
i
SS
1

i
S
nS
dV
dSn

1
V
F
r

S
r

F
0r
SFrrr


3
A function φ harmonic in a volume
V can be expressed in terms of the
function and its normal derivative
on the surface S bounding V.

 






















S SFSF
F
dS
rrnnrr
T
r
11
4





where
VoutsidendSndS
SonF
VinF
T







 11
2
1
1
General Three Dimensional Complex function
If φ (z) is analytic inside and on a
simple closed curve C and a is any
point inside C then





C
dz
az
z
i
a



2
1
Cauchy’s Integral Formula
C
x
y
R
a

see
Vector Analysis.ppt

288. 288
SOLO Vector Analysis
Harmonic Functions (continue 2)
A continuous function φ with continuous first and second partial derivatives is said
to be harmonic if it satisfies Laplace’s Equation 0
2

Properties of Harmonic Functions (continue – 2)
R
S
dSn

1
V
Fr

Sr

F
SF
rrR


4
If the surface S is a sphere S
R
of radius R
with center at then



RS
RF
dS
R
r 


2
4
1
Fr

If f (z) is analytic inside and
on a circle C of radius r and
center at z = a, then
Complex functionGeneral Three Dimensional


 














2
0
2
1
2
1
dera
dz
az
z
i
a
i
eraz
C
i
C
x
y
a
r
see
Vector Analysis.ppt

289. 289
SOLO Vector Analysis
Harmonic Functions (continue 3)
A continuous function φ with continuous first and second partial derivatives is said
to be harmonic if it satisfies Laplace’s Equation 0
2

Properties of Harmonic Functions (continue – 3)
RS
dSn

1
V
F
r

Sr

F
SF
rrR


5 If φ is harmonic in a volume V bounded
by the surface S and if φ = c = constant
at every point on S, then φ = c at every point
of V.
Complex functionGeneral Three Dimensional
Gauss’ Mean Value Theorem
If φ (z) is analytic inside and
on a closed curve C and
φ (z) =c =constant at every
point on C, then φ (a) = c at
every point inside C, i.e.,


Cinsidezcd
c
dz
azi
c
dz
az
z
i
a
i
eraz
CC
















2
0
2
1
22
1
C
x
y
R
a

If φ is harmonic in a region V
bounded by a surface S and ∂ φ/∂ n = 0
at every point of S, then φ = constant at
every point of V.
6
see
Vector Analysis.ppt

290. 290
SOLO Vector Analysis
Harmonic Functions (continue 4)
A continuous function φ with continuous first and second partial derivatives is said
to be harmonic if it satisfies Laplace’s Equation 0
2

Properties of Harmonic Functions (continue – 4)
A non-constant function φ
harmonic in a region V can have
neither a maximum nor a minimum
in V.
S
dSn

1
V
Fr

S
r

SF
rrr

 SF
7
Maximum Modulus Theorem
If f (z) is analytic inside and on a
simple closed curve C and is not
identically equal to a constant, then
the maximum value of | f (z) | occurs
on C.
Complex functionGeneral Three Dimensional
Minimum Modulus Theorem
If f (z) is analytic inside and on a
simple closed curve C and f (z) ≠ 0
inside C then | f (z) | assumes its
minimum value on C.
see
Vector Analysis.ppt

291. 291
SOLO Vector Analysis
Harmonic Functions (continue 5)
A continuous function φ with continuous first and second partial derivatives is said
to be harmonic if it satisfies Laplace’s Equation 0
2

Properties of Harmonic Functions (continue – 5)
8
If φ
1 and φ
2 are two solutions of
Laplace’s equation in a volume V
whose normal derivatives take the
same value ∂ φ
1
/∂ n = ∂ φ
2
/∂ n on the
surface S bounding V, then φ
1
and φ
2

can differ only by a constant.
S
dSn

1
V
Fr

Sr

F
SFrrr


0


S
n

Table of Contents
Complex functionGeneral Three Dimensional
If φ
1
and φ
2
are two analytic functions
inside a curve C whose derivatives take
the same value ∂ φ1/∂ n = ∂ φ2/∂ n on
C, then φ1 and φ2 can differ only by a
constant.










C
C
dz
az
z
i
a
dz
az
z
i
a
'
2
1
'
'
2
1
'
2
2
1
1






  Cinsideaaa ''
21

 Cinsideaconstaa 
21
 
Cona
zz

''
21

see
Vector Analysis.ppt

292. 292
SOLO
Complex Variables
Blaschke Products
Wilhelm Johann Eugen
Blaschke
(1885 - 1962,)
A sequence of points (a
n
) inside the unit disk is said to satisfy the
Blaschke Condition when
Given a sequence obeying the Blaschke Condition, the Blaschke
Product is defined as
provided a
n
≠ 0. Here a
n
*
is the complex conjugate of a
n
. When a
n
= 0 take B(0,z) = z.
The Blaschke Product B(z) defines a function analytic in the open unit disc, and
zero exactly at the a
n
(with multiplicity counted): furthermore it is in the Hardy
class H
∞
.
The sequence of a
n
satisfying the convergence criterion above is sometimes
called a Blaschke Sequence.
 
n
n
a1






n
znB
n
n
n
n
za
za
a
a
zB

,
*1