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1、ComplexAnalysisGeorgeCain(c)Copyright1999byGeorgeCain.Allrightsreserved.TableofContentsChapterOne-ComplexNumbers1.1Introduction1.2Geometry1.3PolarcoordinatesChapterTwo-ComplexFunctions2.1Functionsofarealvariable2.2Functionsofacomplexvariable2.3DerivativesChapterThree-Elem
2、entaryFunctions3.1Introduction3.2Theexponentialfunction3.3Trigonometricfunctions3.4LogarithmsandcomplexexponentsChapterFour-Integration4.1Introduction4.2Evaluatingintegrals4.3AntiderivativesChapterFive-Cauchy'sTheorem5.1Homotopy5.2Cauchy'sTheoremChapterSix-MoreIntegration
3、6.1Cauchy'sIntegralFormula6.2Functionsdefinedbyintegrals6.3Liouville'sTheorem6.4MaximummoduliChapterSeven-HarmonicFunctions7.1TheLaplaceequation7.2Harmonicfunctions7.3Poisson'sintegralformulaChapterEight-Series8.1Sequences8.2Series8.3Powerseries8.4Integrationofpowerseries
4、8.5DifferentiationofpowerseriesChapterNine-TaylorandLaurentSeries9.1Taylorseries9.2LaurentseriesChapterTen-Poles,Residues,andAllThat10.1Residues10.2PolesandothersingularitiesChapterEleven-ArgumentPrinciple11.1Argumentprinciple11.2Rouche'sTheorem---------------------------
5、-------------------------------------------------GeorgeCainSchoolofMathematicsGeorgiaInstituteofTechnologyAtlanta,Georgia0332-0160cain@math.gatech.eduChapterOneComplexNumbers1.1Introduction.Letusharkbacktothefirstgradewhentheonlynumbersyouknewweretheordinaryeverydayintege
6、rs.Youhadnotroublesolvingproblemsinwhichyouwere,forinstance,askedtofindanumberxsuchthat3x?6.Youwerequicktoanswer”2”.Then,inthesecondgrade,MissHoltaskedyoutofindanumberxsuchthat3x?8.Youwerestumped—therewasnosuch”number”!YouperhapsexplainedtoMissHoltthat3?2??6and3?3??9,ands
7、ince8isbetween6and9,youwouldsomehowneedanumberbetween2and3,butthereisn’tanysuchnumber.Thuswereyouintroducedto”fractions.”Thesefractions,orrationalnumbers,weredefinedbyMissHolttobeorderedpairsofintegers—thus,forinstance,?8,3?isarationalnumber.Tworationalnumbers?n,m?and?p,q
8、?weredefinedtobeequalwhenevernq?pm.(Moreprecisely,inotherwords,arationalnumberisanequivalencecla
