All Chapters Covered
M M
SOLUTION MANUAL
M
,Contents
1 Complex Numbers M 1
2 BasicM AlgebraicM Properties 1
3 FurtherM Properties 2
4 VectorsM andM Moduli 3
5 ComplexM Conjugates 5
8M ArgumentsM ofMProductsM andM Quotients 8
10 Examples 12
11 RegionsM inM theM ComplexM Plane 18
2 Analytic Functions
M 22
12 FunctionsM ofMaM ComplexM Variable 22
18MContinuity 22
20M DifferentiationM Formulas 24
23M PolarMCoordinates 25
25 Examples 31
26 HarmonicM Functions 32
3 Elementary FunctionsM 35
29M TheM ExponentialM Function 35
31 BranchesM andM DerivativesM ofMLogarithms 39
32 SomeM IdentitiesM InvolvingM Logarithms 41
33 ComplexM Exponents 43
34 TrigonometricM Functions 45
35 HyperbolicM Functions 49
iii
,iv
4 Integrals 53
38MDefiniteMIntegralsMofMFunctionsM w(t) 53
39MContours 54
42MExamplesMwithMBranchMCuts 56
43M UpperMBoundsM forMModuliM ofMContourMIntegrals 61
45M ProofMofMtheM Theorem 65
49MMultiplyMConnectedMDomains 66
52MSomeMConsequences MofMtheMExtension 69
5 Series 75
56M ConvergenceM ofMSeries 75
59MExamples 77
62M Examples 80
66MUniquenessMofMSeriesM Representations 86
67M MultiplicationM andMDivisionMofMPowerMSeries 89
6 Residues and Poles
M M 94
71M ResidueM atMInfinity 94
72M TheMThreeMTypesMofMIsolatedMSingularMPoints 99
74M Examples 103
76MZerosM andMPoles 109
7 Applications of Residues
M M 118
79M Example 118
81M Jordan'sM Lemma 129
, V
84M IntegrationM AlongM aMBranchM Cut 138
85M DefiniteM IntegralsM InvolvingM SinesM andM Cosines 151
87M Rouch~'sM Theorem 153
89MExamples 155
NoteM toM theM reader:
TheM numberingM systemMusedMhereMtoM identifyMchapters,M sections,M andMexercisesM isMconsistentM
withMthatM usedM inM theM text.M ForMinstance,M accordingM toM theM tableM ofMcontentsMjustM above,M so
lutionsM ofMexercisesM followingM SectionM 10M inM ChapterM 1M ofMtheM textMstartM onM pageM 12M ofMthis
M solutionsM manual.
M M
SOLUTION MANUAL
M
,Contents
1 Complex Numbers M 1
2 BasicM AlgebraicM Properties 1
3 FurtherM Properties 2
4 VectorsM andM Moduli 3
5 ComplexM Conjugates 5
8M ArgumentsM ofMProductsM andM Quotients 8
10 Examples 12
11 RegionsM inM theM ComplexM Plane 18
2 Analytic Functions
M 22
12 FunctionsM ofMaM ComplexM Variable 22
18MContinuity 22
20M DifferentiationM Formulas 24
23M PolarMCoordinates 25
25 Examples 31
26 HarmonicM Functions 32
3 Elementary FunctionsM 35
29M TheM ExponentialM Function 35
31 BranchesM andM DerivativesM ofMLogarithms 39
32 SomeM IdentitiesM InvolvingM Logarithms 41
33 ComplexM Exponents 43
34 TrigonometricM Functions 45
35 HyperbolicM Functions 49
iii
,iv
4 Integrals 53
38MDefiniteMIntegralsMofMFunctionsM w(t) 53
39MContours 54
42MExamplesMwithMBranchMCuts 56
43M UpperMBoundsM forMModuliM ofMContourMIntegrals 61
45M ProofMofMtheM Theorem 65
49MMultiplyMConnectedMDomains 66
52MSomeMConsequences MofMtheMExtension 69
5 Series 75
56M ConvergenceM ofMSeries 75
59MExamples 77
62M Examples 80
66MUniquenessMofMSeriesM Representations 86
67M MultiplicationM andMDivisionMofMPowerMSeries 89
6 Residues and Poles
M M 94
71M ResidueM atMInfinity 94
72M TheMThreeMTypesMofMIsolatedMSingularMPoints 99
74M Examples 103
76MZerosM andMPoles 109
7 Applications of Residues
M M 118
79M Example 118
81M Jordan'sM Lemma 129
, V
84M IntegrationM AlongM aMBranchM Cut 138
85M DefiniteM IntegralsM InvolvingM SinesM andM Cosines 151
87M Rouch~'sM Theorem 153
89MExamples 155
NoteM toM theM reader:
TheM numberingM systemMusedMhereMtoM identifyMchapters,M sections,M andMexercisesM isMconsistentM
withMthatM usedM inM theM text.M ForMinstance,M accordingM toM theM tableM ofMcontentsMjustM above,M so
lutionsM ofMexercisesM followingM SectionM 10M inM ChapterM 1M ofMtheM textMstartM onM pageM 12M ofMthis
M solutionsM manual.
