Functions of a Complex Variable I Math 561, Fall 2021 Jens Lorenz September 8, 2021 Department of Mathematics and Statistics, UNM, Albuquerque, NM 87131 Contents 1 The Field C of Complex Numbers; Some Simple Concepts 6 1.1 The Field C of Complex Numbers and the Euclidean Plane . .6 1.2 Some Simple Concepts . .8 1.3 Complex Differentiability . 13 1.4 Alternating Series . 14 1.5 Historical Remarks . 16 2 The Cauchy Product of Two Series: Proof of the Addition Theorem for the Exponential Function 17 2.1 The Cauchy Product of Two Series . 17 2.2 The Addition Theorem for the Exponential Function . 20 2.3 Powers of e ......................................... 21 2.3.1 Integer Powers . 21 2.3.2 Rational Powers . 22 2.4 Euler's Identity and Implications . 23 2.5 The Polar Representation of a Complex Number . 24 2.6 Further Properties of the Exponential Function . 26 2.7 The Main Branch of the Complex Logarithm . 26 2.8 Remarks on the Multivalued Logarithm . 29 3 Complex Differentiability 30 3.1 Outline and Notations . 30 3.2 R{Linear and C{Linear Maps from R2 C into Itself . 30 3.3 The Polar Representation of a Complex' Number and the Corresponding Matrix . 31 3.4 Real and Complex Differentiability . 32 3.5 The Complex Logarithm as an Example . 34 3.6 The Operators @=@z and @=@z¯ .............................. 36 1 4 Complex Line Integrals and Cauchy's Theorem 39 4.1 Curves . 39 4.2 Definition and Simple Properties of Line Integrals . 40 4.3 Goursat's Lemma . 44 4.4 Construction of a Primitive in a Disk . 46 4.5 Cauchy's Theorem in a Disk . 48 4.6 Extensions . 48 4.7 Cauchy's Integral Formula in a Disk . 51 5 Holomorphic Functions Written As Power Series 55 5.1 Main Result . 55 5.2 The Geometric Series . 56 5.3 Power Series Expansion Using the Geometric Series . 57 6 Functions Defined by Power Series 60 6.1 Remarks on the Exchange of Limits . 61 6.2 The Disk of Convergence of a Power Series . 62 6.3 Remarks on lim sup and lim inf . 65 6.4 The Radius of Convergence of a Power Series: Hadamard's Formula . 68 6.5 Matrix{Valued Analytic Functions and Hadamard's Formula for the Spectral Radius of a Matrix . 70 6.6 Differentiation of Power Series . 71 7 The Cauchy Estimates and Implications 77 7.1 The Cauchy Estimates . 77 7.2 Liouville's Theorem . 77 7.3 The Fundamental Theorem of Algebra . 78 7.4 The Zeros of p(z) and p0(z)................................ 80 8 Morera's Theorem and Locally Uniform Limits of Holomorphic Functions; Stir- ling's Formula for the Γ{Function 82 8.1 On Connected Sets . 82 8.2 Morera's Theorem . 82 8.3 Modes of Convergence of a Sequence of Functions . 83 8.4 Integration with Respect to a Parameter . 86 8.5 Application to the Γ–Function: Analyticity in the Right Half{Plane . 87 8.6 Stirling's Formula . 89 9 Zeros of Holomorphic Functions and the Identity Theorem; Analytic Continua- tion 94 10 Isolated Singularities and Laurent Expansion 96 10.1 Classification of Isolated Singularities . 96 10.2 Removable Singularities . 97 10.3 Theorem of Casorati{Weierstrass on Essential Singularities . 98 10.4 Laurent Series . 100 10.4.1 Terminology . 100 10.4.2 Characterization of Isolated Singularities in Terms of Laurent Expansions . 101 10.4.3 Convergence of Laurent Series . 101 2 10.4.4 Examples . 102 10.4.5 Laurent Expansion: Uniqueness . 103 10.4.6 Laurent Expansion: Existence . 104 10.4.7 Local Behavior and Laurent Expansion . 106 11 The Calculus of Residues; Evaluation of Integrals; Partial Fraction Decomposi- tions 108 11.1 Computation of Residues . 108 11.1.1 The Case of a Simple Pole . 109 11.1.2 Poles of Order k .................................. 110 11.2 Calculus of Residues . 113 11.2.1 Direct Applications of the Residue Theorem . 113 11.2.2 Substitution of z = eit ............................... 115 11.2.3 Integrals over < x < ............................ 116 11.2.4 Extensions Using−∞ Jordan's1 Lemma . 118 11.2.5 A Pole on the Real Axis . 120 11.2.6 Use of a Second Path . 123 11.3 Derivation of a Partial Fraction Decomposition via Integration . 125 11.4 The Partial Fraction Decomposition of π cot(πz).................... 128 11.5 Summary of Examples . 130 11.6 Practice Problems . 131 12 The Bernoulli Numbers, the Values ζ(2m), and Sums of Powers 133 12.1 The Bernoulli Numbers . 133 12.2 The Taylor Series of z cot z in Terms of Bernoulli Numbers . 136 12.3 The Values ζ(2m); m = 1; 2;::: .............................. 136 12.4 Sums of Powers and Bernoulli Numbers . 138 13 Properties of the Γ{Function 141 13.1 Extension of the Domain of Definition of Γ(z) Using the Functional Equation . 141 13.2 Extension of the Domain of Definition of Γ(z) Using Series Expansion . 142 13.3 The Reflection Formula . 143 13.4 Special Values of Γ(z)................................... 144 13.5 Simple Applications . 145 13.6 The Function ∆(z) = 1=Γ(z)............................... 146 13.7 Log{Convexity of Γ(x)................................... 146 13.8 Summary . 148 14 Log Functions 149 14.1 The Main Branch of log(z): First Introduction . 149 14.2 Auxiliary Results . 149 14.3 The Main Branch of the Complex Logarithm: Inversion of w ew .......... 150 14.4 Complex Logarithms in Other Simply Connected Regions . .! . 152 14.5 Argument Functions . 153 15 Extensions of Cauchy's Theorem in a Disk 155 15.1 Homotopic Curves . 155 15.2 Cauchy's Theorem . 155 3 16 The General Residue Theorem and the Argument Principle 158 16.1 Remarks on Solutions of Equations under Perturbations . 158 16.2 The Winding Number or Index . 159 16.3 The General Residue Theorem . 160 16.4 Zero{Counting of Holomorphic Maps . 162 16.4.1 The Multiplicity of a Zero . 162 16.4.2 The Zeros of a Holomorphic Function in a Disk . 162 16.4.3 The Argument Principle and Log{Functions . 164 16.5 The Change of Argument and Zeros of Polynomials . 167 17 Applications and Extensions of the Argument Principle 170 17.1 Perturbation of an Equation: An Example . 170 17.2 Perturbation of a Multiple Zero . 171 17.3 The Open Mapping Theorem . 173 17.4 Extension of Theorem 16.2 . 174 17.5 Local Inverses of Holomorphic Functions . 174 17.6 The Argument Principle for Meromorphic Functions . 178 17.7 Rouch´e'sTheorem and Hurwitz's Theorem . 179 17.8 An Application of Rouch´e'sTheorem . 181 18 Matrix{Valued and Operator Valued Analytic Functions 182 18.1 Outline and Examples . 182 18.2 Analyticity of the Resolvent . ..