Taylor’s series

1. taylor’s series

2. taylor’s series
If f(z) is a complex function analytic inside & on a simple
closed curve C (usually a circle) in the z-plane, then the higher
derivatives of f(z) also exist inside C. If z
0& z
0+h are two
fixed points inside C then…..
F(z
0+h) = f(z
0) + hf
(1)
(z
0) + h
2
/2! f
(2)
(z
0)
+…..+ h
n
/n! f
(n)
(z
0)+….
Where f
(k)
(z
0) is the k
th
derivative of f(z) at z=z
0

3. taylor’s series
Putting z
0+h=z, the series becomes….
F(z) = f(z
0) + (z-z
0) f
(1)
(z
0) + (z-z
0)
2
/2! f
(2)
(z
0) +
…..+ (z-z
0)
n
/n! f
n
(z
0) +…….
= f
(n)
(z
0).
This series is known as Taylor’s series expansion of f(z) about
z=z
0.The radius of converges of this series is | z –z
0| < R, a
disk centered on z=z
0& of radius R.

4. taylor’s series
When z
0=0, in previous equation then we get,
F(z) = f(0) + zf
(1)
(0) + z
2
/2! f
(2)
(0) +…..+ z
n
/n!
f
(n)
(0) +…..
The obtained series is Maclaurin’s Seriesexpansion, In the
case of function of real variables.

5. taylor’s series
Some Standard power series expansion :-
1. ; | z | < ∞
2. ; | z | < ∞
3. ; | z | < ∞

6. taylor’s series
4. ; | z | < ∞
5. ; | z | < ∞
6.(Geometric Series)

7. taylor’s series
7. ; | z | < 1
; | z | < 1
8. ( Binomial series for any positive integer m )
; | z | < 1

8. taylor’s series
Illustration 5.2: Find the Taylor’s series expansion of
f(z)=a/(bz+c) about z=z
0. Also determine about the region of
convergence.
Illustration 5.3: Determine the Taylor’s series expansion of the
function f(z)=1/z( z -2i ) above the point z=i.
(a)directly upto the term (z –i)
4
, (b) using the binomial
expansion. Also determine about the radius of convergence.

9. taylor’s series
Illustration 5.4: Find the Taylor’s series expansion of f(z) = 1/( z
2
-z –6 ) about z=1.
Illustration 5.5: Expand f(z) = (z -1)/(z + 1) as Taylor’s series
(a) about the point z=0 , and (b) about the point z=1.
determine also the result the radius of convergence.

10. taylor’s series
Illustration 5.6: (Geometric series) Expand f(z) = 1/(1 –z).
Illustration 5.7: Find the Maclaurin’s series of ln[(1 + z)/(1 –z)].
Illustration 5.8: Find the Maclaurin’s series of f(z) = 1/(1 –z
2
).

11. taylor’s series
Illustration 5.9: Find the Maclaurin’s series of f(Z) = tan
-1
z.
Illustration 5.10: (Development by using the geometric series)
Develop 1/(c –z) in powers of z –z
0, where c –z
0≠ 0.
Illustration 5.11: (Reduction by partial fractions) Find the
Taylor’s series of f(z)= with center z
0= 1.

12. taylor’s series
Illustration 5.12: Develop f(z) = sin
2
z in a Maclaurin’s series and
find the radius of convergence.
Note: This example can also be solved using the formula
sin
2
z = ( 1 –cos2z) / 2 & then using the standard power series
expansion for cos z with z replaced by 2z.

13. taylor’s series
In the above discussion of power series and in particular
Taylor’s series with illustrations, we have seen that inside the
radius of convergence, the given function and its Taylor’s
series expansion are identically equal. Now the points at which
a function fails to be analytic are called Singularities.
No Taylor’s series exp. Is possible about singularity.
So, Taylor’s exp. About a point z
0, at which a function is
analytic is only valid within a circle centered z
0.
Thus all the singularities must be excluded in Taylor’s
Expansion.

14. Thank you
Prepared By
BHARGAV GODHANI