All Chapters Covered
SOLUTION MANUAL
,Contents
1 Complex Numbers 1
2 Basic Algebraic Properties 1
3 Further Properties 2
4 Vectors and Moduli 3
5 Complex Conjugates 5
8 Arguments of Products and Quotients 8
10 Examples 12
11 Regions in the Complex Plane 18
2 Analytic Functions 22
12 Functions of a Complex Variable 22
18 Continuity 22
20 Differentiation Formulas 24
23 Polar Coordinates 25
25 Examples 31
26 Harmonic Functions 32
3 Elementary Functions 35
29 The Exponential Function 35
31 Branches and Derivatives of Logarithms 39
32 Some Identities Involving Logarithms 41
33 Complex Exponents 43
34 Trigonometric Functions 45
35 Hyperbolic Functions 49
iii
,iv
4 Integrals 53
38 Definite Integrals of Functions w(t) 53
39 Contours 54
42 Examples with Branch Cuts 56
43 Upper Bounds for Moduli of Contour Integrals 61
45 Proof of the Theorem 65
49 Multiply Connected Domains 66
52 Some Consequences of the Extension 69
5 Series 75
56 Convergence of Series 75
59 Examples 77
62 Examples 80
66 Uniqueness of Series Representations 86
67 Multiplication and Division of Power Series 89
6 Residues and Poles 94
71 Residue at Infinity 94
72 The Three Types of Isolated Singular Points 99
74 Examples 103
76 Zeros and Poles 109
7 Applications of Residues 118
79 Example 118
81 Jordan's Lemma 129
, V
84 Integration Along a Branch Cut 138
85 Definite Integrals Involving Sines and Cosines 151
87 Rouch~'s Theorem 153
89 Examples 155
Note to the reader:
The numbering system used here to identify chapters, sections, and exercises is consistent with
that used in the text. For instance, according to the table of contents just above, solutions of
exercises following Section 10 in Chapter 1 of the text start on page 12 of this solutions manual.
SOLUTION MANUAL
,Contents
1 Complex Numbers 1
2 Basic Algebraic Properties 1
3 Further Properties 2
4 Vectors and Moduli 3
5 Complex Conjugates 5
8 Arguments of Products and Quotients 8
10 Examples 12
11 Regions in the Complex Plane 18
2 Analytic Functions 22
12 Functions of a Complex Variable 22
18 Continuity 22
20 Differentiation Formulas 24
23 Polar Coordinates 25
25 Examples 31
26 Harmonic Functions 32
3 Elementary Functions 35
29 The Exponential Function 35
31 Branches and Derivatives of Logarithms 39
32 Some Identities Involving Logarithms 41
33 Complex Exponents 43
34 Trigonometric Functions 45
35 Hyperbolic Functions 49
iii
,iv
4 Integrals 53
38 Definite Integrals of Functions w(t) 53
39 Contours 54
42 Examples with Branch Cuts 56
43 Upper Bounds for Moduli of Contour Integrals 61
45 Proof of the Theorem 65
49 Multiply Connected Domains 66
52 Some Consequences of the Extension 69
5 Series 75
56 Convergence of Series 75
59 Examples 77
62 Examples 80
66 Uniqueness of Series Representations 86
67 Multiplication and Division of Power Series 89
6 Residues and Poles 94
71 Residue at Infinity 94
72 The Three Types of Isolated Singular Points 99
74 Examples 103
76 Zeros and Poles 109
7 Applications of Residues 118
79 Example 118
81 Jordan's Lemma 129
, V
84 Integration Along a Branch Cut 138
85 Definite Integrals Involving Sines and Cosines 151
87 Rouch~'s Theorem 153
89 Examples 155
Note to the reader:
The numbering system used here to identify chapters, sections, and exercises is consistent with
that used in the text. For instance, according to the table of contents just above, solutions of
exercises following Section 10 in Chapter 1 of the text start on page 12 of this solutions manual.
