multivariate-CalculusCHAP1_Complex_Analysis.pdf

Multivariate calculator

1. Department of Electronic Engineering
The IslamiaUniversity of Bahawalpur, Pakistan
Complex Numbers
MATH-00208 Multivariate Calculus & Complex Analysis

2. 2
Chapter Outline
1.1 -The Real Number System
1.2 -Graphical Representation of Real Numbers
1.3 -The Complex Number System
1.4 -Fundamental Operations with Complex Numbers
1.5 -Absolute Value
1.6 -Axiomatic Foundation of the Complex Number System
1.7 -Graphical Representation of Complex Numbers
1.8 -Polar Form of Complex Numbers
1.9 -De Moivre’sTheorem
1.10 -Roots of Complex Numbers
1.11 -Euler’s Formula
1.12 -Polynomial Equations
1.13 -The nth Roots of Unity
1.14 -Vector Interpretation of Complex Numbers
1.15 -Stereographic Projection
1.16 -Dot and Cross Product
1.17 -Complex Conjugate Coordinates

3. 3
1.1 The Real Number System
Thenumbersystemasweknowittodayisaresultofgradual
developmentasindicatedinthefollowinglist.
1)Naturalnumbers1,2,3,4,...,alsocalledpositiveintegers,werefirst
usedincounting.
Ifaandbarenaturalnumbers,thesuma+bandproducta‧b,(a)(b)
orabarealsonaturalnumbers.
Forthisreason,thesetofnaturalnumbersissaidtobeclosedunder
theoperationsofadditionandmultiplicationortosatisfytheclosure
propertywithrespecttotheseoperations.
2)Negativeintegersandzero,denotedby-1,-2,-3,...and0,
respectively,permitsolutionsofequationssuchasx+b=awherea
andbareanynaturalnumbers.
Thisleadstotheoperationofsubtraction,orinverseofaddition,and
wewritex=a-b.
Thesetofpositiveandnegativeintegersandzeroiscalledthesetof
integersandisclosedundertheoperationsofaddition,multiplication,
andsubtraction.

4. 4
1.1 The Real Number System
3)Rationalnumbersorfractionssuchas3/4,-8/3,...permitsolutionsof
equationssuchasbx=aforallintegersaandbwhereb≠0.
Thisleadstotheoperationofdivisionorinverseofmultiplication,and
wewritex=a/bora
¼
b(calledthequotientofaandb)whereais
thenumeratorandbisthedenominator.
Thesetofintegersisapartorsubsetoftherationalnumbers,since
integerscorrespondtorationalnumbersa/bwhereb=1.
Thesetofrationalnumbersisclosedundertheoperationsofaddition,
subtraction,multiplication,anddivision,solongasdivisionbyzerois
excluded.
4)Irrationalnumberssuchas
 
andarenumbersthatcannotbe
expressedasa/bwhereaandbareintegersandb≠0.
Thesetofrationalandirrationalnumbersiscalledthesetofreal
numbers.

5. 5
1.2 Graphical Representation of Real Numbers
Realnumberscanberepresentedbypointsonalinecalledthereal
axis,asindicatedinFig.1-1.
Thepointcorrespondingtozeroiscalledtheorigin.
Conversely,toeachpointonthelinethereisoneandonlyonereal
number.
IfapointAcorrespondingtoarealnumberaliestotherightofa
pointBcorrespondingtoarealnumberb,wesaythataisgreaterthan
borbislessthanaandwritea>borb<a,respectively.
Thesetofallvaluesofxsuchthata<x<biscalledanopeninterval
ontherealaxiswhilea≤x≤b,whichalsoincludestheendpointsa
andb,iscalledaclosedinterval.

6. 6
1.2 Graphical Representation of Real Numbers
Thesymbolx,whichcanstandforanyrealnumber,iscalledareal
variable.
Theabsolutevalueofarealnumbera,denotedby|a|,isequaltoaifa
>0,to-aifa<0andto0ifa=0.
Thedistancebetweentwopointsaandbontherealaxisis|a–b|.

7. 7
Thereisnorealnumberxthatsatisfiesthepolynomialequationx
2
+1
=0.
Topermitsolutionsofthisandsimilarequations,thesetofcomplex
numbersisintroduced.
Wecanconsideracomplexnumberashavingtheforma+biwherea
andbarerealnumbersandi,whichiscalledtheimaginaryunit,has
thepropertythati
2
=-1.
Ifz=a+bi,thenaiscalledtherealpartofzandbiscalledthe
imaginarypartofzandaredenotedbyRe{z}andIm{z},respectively.
Thesymbolz,whichcanstandforanycomplexnumber,iscalleda
complexvariable.
Twocomplexnumbersa+biandc+diareequalifandonlyifa=c
andb=d.
Wecanconsiderrealnumbersasasubsetofthesetofcomplex
numberswithb=0.
1.3 The Complex Number System

8. 8
Accordinglythecomplexnumbers0+0iand-3+0irepresentthereal
numbers0and-3,respectively.
Ifa=0,thecomplexnumber0+biorbiiscalledapureimaginary
number.
Thecomplexconjugate,orbrieflyconjugate,ofacomplexnumbera
+biisa-bi.
Thecomplexconjugateofacomplexnumberzisoftenindicatedby
orz*.
1.3 The Complex Number System

9. 9
Inperformingoperationswithcomplexnumbers,wecanproceedasin
thealgebraofrealnumbers,replacingi
2
by-1whenitoccurs.
1.4 Fundamental Operations with Complex Numbers

10. 10
Theabsolutevalueormodulusofacomplexnumbera+biisdefined
as
1.5 Absolute Value
Ifz
1
,z
2
,z
3
,...,z
m
arecomplexnumbers,thefollowingpropertieshold.
5
6
5
6

11. 11
Fromastrictlylogicalpointofview,itisdesirabletodefinea
complexnumberasanorderedpair(a,b)ofrealnumbersaandb
subjecttocertainoperationaldefinitions,whichturnouttobe
equivalenttothosedefinedinlasttopics.
Thesedefinitionsareasfollows,wherealllettersrepresentreal
numbers.
1.6 Axiomatic Foundation of the Complex Number System
Fromthesewecanshow[Problem1.14]that
(a, b) = a(1, 0) + b(0, 1)
andweassociatethiswitha+biwhereiisthesymbolfor(0,1)and
hasthepropertythati
2
=(0,1)(0,1)=(-1,0)
and(1,0)canbeconsideredequivalenttotherealnumber1.
Theorderedpair(0,0)correspondstotherealnumber0.

12. 12
1.6 Axiomatic Foundation of the Complex Number System

13. 13
Fromtheabove,wecanprovethefollowing.
1.6 Axiomatic Foundation of the Complex Number System
Ingeneral,anysetsuchasS,whosememberssatisfytheabove,is
calledafield.
0iscalledtheidentitywithrespecttoaddition,1iscalledtheidentitywithrespect
tomultiplication.

14. 14
1.7 Graphical Representation of Complex Numbers
Supposerealscalesarechosenontwomutuallyperpendicularaxes
X΄OXandY΄OY[calledthexandyaxes,respectively]asinFig.1-2.
Wecanlocateanypointintheplanedeterminedbytheselinesbythe
orderedpairofrealnumbers(x,y)calledrectangularcoordinatesof
thepoint.
ExamplesofthelocationofsuchpointsareindicatedbyP,Q,R,S,
andTinFig.1-2.

15. 15
1.7 Graphical Representation of Complex Numbers
Sinceacomplexnumberx+iycanbeconsideredasanorderedpairof
realnumbers,wecanrepresentsuchnumbersbypointsinanxyplane
calledthecomplexplaneorArganddiagram.
ThecomplexnumberrepresentedbyP,forexample,couldthenbe
readaseither(3,4)or3+4i.
Toeachcomplexnumbertherecorrespondsoneandonlyonepointin
theplane,andconverselytoeachpointintheplanetherecorresponds
oneandonlyonecomplexnumber.
Becauseofthisweoftenrefertothecomplexnumberzasthepointz.
Sometimes,werefertothexandyaxesastherealandimaginary
axes,respectively,andtothecomplexplaneasthezplane.
Thedistancebetweentwopoints,z
1
=x
1
+iy
1
andz
2
=x
2
+iy
2
,inthe
complexplaneisgivenby

16. 16
1.8 Polar Form of Complex Numbers
LetPbeapointinthecomplexplanecorrespondingtothecomplex
number(x,y)orx+iy.
ThenweseefromFig.1-3that
where
 
iscalledthe
modulusorabsolutevalueofz=x+iy
[denotedbymodzor|z|]
θ,calledtheamplitudeorargumentof
z=x+iy[denotedbyargz],isthe
anglethatlineOPmakeswiththe
positivexaxis.
whichiscalledthepolarformofthecomplexnumber,andrandθare
calledpolarcoordinates.
Itfollowsthat

17. 17
1.8 Polar Form of Complex Numbers
Itissometimesconvenienttowritetheabbreviationcisθforcosθ+
isinθ.
Foranycomplexnumberz≠0therecorrespondsonlyonevalueofθ
in0≤θ≤2π.
However,anyotherintervaloflength2π,forexample-π≤θ<πcan
beused.
Anyparticularchoice,decideduponinadvance,iscalledtheprincipal
range,andthevalueofθiscalleditsprincipalvalue.

18. 18
1.9 De Moivre’sTheorem
Let
then we can show that [see Problem 1.19]
A generalization of (1.2) leads to
and if z
1
= z
2
= … = z
n
= zthis becomes
which is often called De Moivre’stheorem.

19. 19
1.9 De Moivre’sTheorem

20. 20
1.10 Roots of Complex Numbers
Anumberwiscalledannthrootofacomplexnumberzifw
n
=z,and
wewritew=z
1/n
.
FromDeMoivre’stheoremwecanshowthatifnisapositiveinteger,
fromwhichitfollowsthattherearendifferentvaluesforz
1/n
,i.e.,n
differentnthrootsofz,providedz≠0.
De Moivre’stheorem

21. 21
1.10 Roots of Complex Numbers
Comparing above two equations

22. 22
1.10 Roots of Complex Numbers
Byconsideringk=5,6,...aswellasnegativevalues,-1,-2,...
repetitionsoftheabovefivevaluesofzareobtained.
Hence,thesearetheonlysolutionsorrootsofthegivenequation.
Thesefiverootsarecalledthefifthrootsof-32andarecollectively
denotedby(-32)
1/5
.
Ingeneral,a
1/n
representsthenthrootsofaandtherearensuchroots.

23. 23
1.11 Euler’s Formula
Byassumingthattheinfiniteseriesexpansion
of elementary calculus holds when x= iθ, we can arrive at the result
whichiscalledEuler’sformula.
Ingeneral,wedefine
Inthespecialcasewherey=0thisreducestoe
x
.
Notethatintermsof(1.7)DeMoivre’stheoremreducesto(e
iθ
)
n
=
e
inθ
.
Itismoreconvenient,however,simplytotake(1.7)asadefinitionof
e
iθ
.
De Moivre’stheorem

24. 24
1.12 Polynomial Equations
Ofteninpracticewerequiresolutionsofpolynomialequationshaving
theform
Suchsolutionsarealsocalledzerosofthepolynomialontheleftof
(1.9)orrootsoftheequation.
Averyimportanttheoremcalledthefundamentaltheoremofalgebra
statesthateverypolynomialequationoftheform(1.9)hasatleastone
rootinthecomplexplane.
Fromthiswecanshowthatithasinfactncomplexroots,someorall
ofwhichmaybeidentical.
Ifz
1
,z
2
,...,z
n
arethenroots,then(1.9)canbewritten
wherea
0
≠0,a
1
,...,a
n
aregivencomplexnumbersandnisapositive
integercalledthedegreeoftheequation.
whichiscalledthefactoredformofthepolynomialequation.

25. 25
1.13 The nth Roots of Unity
Thesolutionsoftheequationz
n
=1wherenisapositiveintegerare
calledthenthrootsofunityandaregivenby
Ifwelet
Geometrically,theyrepresentthenverticesofaregularpolygonofn
sidesinscribedinacircleofradiusonewithcenterattheorigin.
Thiscirclehastheequation|z|=1andisoftencalledtheunitcircle.
thenrootsare1,ω,ω
2
,...,ω
n-1
.

26. 26
1.14 Vector Interpretation of Complex Numbers
Acomplexnumberz=x+iycanbeconsideredasavectorOP
whoseinitialpointistheoriginOandwhoseterminalpointPisthe
point(x,y)asinFig.1-4.
WesometimescallOP=x+iythepositionvectorofP.
Twovectorshavingthesamelengthormagnitudeanddirectionbut
differentinitialpoints,suchasOPandABinFig.1-4,areconsidered
equal.
HencewewriteOP=AB=x+iy.

27. 27
1.14 Vector Interpretation of Complex Numbers
Additionofcomplexnumberscorrespondstotheparallelogramlaw
foradditionofvectors[seeFig.1-5].
Thustoaddthecomplexnumbers z
1
andz
2
,wecompletethe
parallelogramOABCwhosesidesOAandOCcorrespondtoz
1
andz
2
.
ThediagonalOBofthisparallelogramcorrespondstoz
1
+z
2
.See
Problem1.5.

28. 28
1.14 Vector Interpretation of Complex Numbers

29. 29
1.15 Stereographic Projection
Letƿ[Fig.1-6]bethecomplexplaneandconsiderasphereStangent
toPatz=0.
ThediameterNSisperpendiculartoƿandwecallpointsNandSthe
northandsouthpolesofS.
CorrespondingtoanypointAonƿwecanconstructlineNA
intersectingSatpointA΄.
Thustoeachpointofthecomplexplaneƿtherecorrespondsoneand
onlyonepointofthesphereS,andwecanrepresentanycomplex
numberbyapointonthesphere.

30. 30
1.15 Stereographic Projection
ForcompletenesswesaythatthepointNitselfcorrespondstothe
“pointatinfinity”oftheplane.
Thesetofallpointsofthecomplexplaneincludingthepointat
infinityiscalledtheentirecomplexplane,theentirezplane,orthe
extendedcomplexplane.
Theabovemethodformappingtheplaneontothesphereiscalled
stereographicprojection.
ThesphereissometimescalledtheRiemannsphere.
WhenthediameteroftheRiemannsphereischosentobeunity,the
equatorcorrespondstotheunitcircleofthecomplexplane.

31. 31
1.16 Dot and Cross Product
Letz
1
=x
1
+iy
1
andz
2
=x
2
+iy
2
betwocomplexnumbers[vectors].
Thedotproduct[alsocalledthescalarproduct]ofz
1
andz
2
isdefined
astherealnumber
whereθistheanglebetweenz
1
andz
2
whichliesbetween0andπ.
Thecrossproductofz
1
andz
2
isdefinedasthevectorz
1
z
2
=(0,0,
x
1
y
2
-y
1
x
2
)perpendiculartothecomplexplanehavingmagnitude
1)Anecessaryandsufficientconditionthat z
1
andz
2
be
perpendicularisthatz
1
‧z
2
=0.
2)Anecessaryandsufficientconditionthatz
1
andz
2
beparallelis
that|z
1
z
2
|=0.
3)Themagnitudeoftheprojectionofz
1
onz
2
is|z
1
‧z
2
|/|z
2
|.
4)Theareaofaparallelogramhavingsidesz
1
andz
2
is|z
1
z
2
|.

32. 32
1.17 Complex Conjugate Coordinates
Apointinthecomplexplanecanbelocatedbyrectangular
coordinates(x,y)orpolarcoordinates(r,θ).
Manyotherpossibilitiesexist.
Onesuchpossibilityusesthefactthat
Thecoordinates(z,)thatlocateapointarecalledcomplex
conjugatecoordinatesorbrieflyconjugatecoordinatesofthepoint
[seeProblem1.43].
where

33. 33
1.17 Complex Conjugate Coordinates