Complex Variable Outline Richard Koch March 16, 2016 Contents List of Figures5 1 Preface 8 2 Preliminaries and Examples 10 2.1 Review of Complex Numbers.......................... 10 2.2 Holomorphic Functions.............................. 11 2.3 Extending Functions from R to C ....................... 14 2.4 Exponential Functions.............................. 15 2.5 Trigonometric Functions............................. 17 2.6 ez as a Map from C to C ............................ 20 2.7 Log(z)....................................... 20 2.8 ab ......................................... 22 3 The Cauchy-Riemann Equations and Consequences 26 3.1 Differentiation Rules............................... 26 3.1.1 The Product Rule............................ 26 3.1.2 The Chain Rule............................. 26 3.1.3 The Inverse Function Rule........................ 26 3.2 Complex Functions as Maps from the Plane to the Plane.......... 26 3.3 The Cauchy-Riemann Equations........................ 27 3.4 The Converse of the Cauchy-Riemann Equations............... 28 3.5 Linear Approximation of Maps R2 ! R2 .................... 29 3.6 The Local Geometry of Holomorphic Maps.................. 30 4 Integration 32 4.1 Definition of the Complex Integral....................... 32 4.2 A Useful Inequality................................ 36 4.3 Consequences of the Inequality......................... 36 1 CONTENTS 2 5 Review of Advanced Calculus 37 5.1 Differentiation.................................. 37 5.2 Geometry of Subsets............................... 37 5.3 Integration.................................... 38 5.4 Fundamental Theorem of Calculus....................... 39 5.5 Differential Forms................................ 39 5.6 Solving gradf = E for f ............................. 40 5.7 Calculus Version of the Residue Theorem................... 43 6 Complex Analysis and Advanced Calculus 44 6.1 Cauchy's Theorem................................ 44 6.2 Power Series Expansions of Holomorphic Functions.............. 46 6.3 Indefinite Complex Integrals........................... 47 6.4 Morera's Theorem................................ 48 6.5 Filling the Gap.................................. 49 6.6 Summary..................................... 51 7 Complex Analysis from Topology 52 7.1 Curves and Homotopy.............................. 52 7.2 Holomorphic f Have Local Antiderivatives................... 55 7.3 Proof of Homotopy Cauchy Theorem...................... 57 7.4 Chapter 6 Done Rigorously........................... 59 8 Winding Numbers, Homology, and Cauchy's Theorem 62 8.1 Winding Numbers................................ 62 8.2 The Inside of a Curve.............................. 63 8.3 The Homological Cauchy Theorem....................... 65 9 Isolated Singularities and the Residue Theorem 70 9.1 The Identity Theorem.............................. 70 9.2 Liouville's Theorem............................... 71 9.3 Isolated Singularities............................... 72 9.4 Newton's Calculus Paper............................ 73 9.5 Laurent Series................................... 74 9.6 Classification of Isolated Singularities...................... 78 9.7 The Residue at an Isolated Singularity..................... 80 9.8 The Residue Theorem.............................. 80 10 Fundamental Results 85 10.1 Meromorphic Functions............................. 85 10.2 The Order of a Zero or Pole........................... 85 10.3 The Riemann Sphere............................... 86 CONTENTS 3 10.4 The Argument Principle............................. 88 10.5 The Open Mapping Theorem.......................... 90 10.6 The Maximum Principle............................. 91 10.7 Uniform Limits of Holomorphic Functions................... 92 11 Automorphism Groups 94 11.1 The Automorphism Group of C ......................... 94 11.2 The Automorphism Group of the Riemann Sphere.............. 95 11.3 The Automorphism Group of the Open Unit Disk.............. 99 11.4 The Unit Disk and Non-Euclidean Geometry................. 101 12 Topology on H(U) 109 12.1 A Metric on H(U) and C(U)........................... 110 12.2 The Key Technical Theorem........................... 111 12.3 Compact Subsets of H(U)............................ 112 13 Riemann Mapping Theorem 114 13.1 The Riemann Mapping Theorem........................ 114 13.2 History of the Riemann Mapping Theorem................... 118 13.3 A Short Sketch of Consequences of the Uniformization Theorem...... 122 13.3.1 Manifolds................................. 122 13.3.2 Riemannian Manifolds.......................... 123 13.3.3 Riemann Surfaces............................ 125 13.3.4 Covering Surface............................. 125 13.3.5 The Uniformization Theorem...................... 126 14 The Theorems of Mittag-Leffler and Weierstrass 127 14.1 The Mittag-Leffler Theorem........................... 127 14.2 Infinite Products................................. 130 14.3 A Convergence Criterion for Products..................... 131 14.4 Infinite Products of Holomorphic Functions.................. 132 14.5 The Theorem of Weierstrauss.......................... 132 15 Classical Sum and Product Formulas 138 15.1 Beautiful Identities................................ 138 15.2 A Product Formula................................ 143 16 The Gamma Function 147 16.1 Three Theorems................................. 147 16.2 Feynman and Differentiating Under the Integral Sign............. 149 16.3 The Gamma Function.............................. 150 16.4 The Gamma Function as an Infinite Product................. 154 CONTENTS 4 16.5 The Two Definitions of Γ(z) Agree....................... 159 17 The Zeta Function 161 17.1 Some History................................... 161 17.2 More History................................... 163 17.3 Dirichlet...................................... 164 17.4 Riemann...................................... 165 17.5 The Riemann Zeta Function........................... 166 17.6 Analytic Continuation of the Zeta Function.................. 168 17.7 The Functional Equation of the Zeta Function................ 171 17.8 Bernoulli Numbers................................ 177 17.9 The Value of the Zeta Function on Integers.................. 179 17.10Remarks on the Prime Number Theorem................... 183 17.11A Crucial Calculation.............................. 186 17.12The Intuitive Meaning of φ(n).......................... 189 17.13Zeros of ζ(z) on the Boundary of the Critical Strip.............. 190 17.14Proof of the Prime Number Theorem...................... 191 17.15Consequences of the Riemann Hypothesis................... 195 17.16Other Results in Riemann's Paper....................... 197 18 Dirichlet Series 200 19 Dirichlet's Theorem 204 19.1 Characters of a Finite Abelian Group..................... 204 19.2 Statement of Dirichlet's Theorem; L Functions................ 206 19.3 Analytic Continuation.............................. 207 19.4 Proof of Theorem Modulo L(1;') 6= 0 for ' 6= 1................ 208 19.5 L(1;') 6= 0 if ' 6= 1................................ 210 1 x 19.6 Proof of π(x; a) ∼ φ(b) ln x ............................ 212 19.7 Final Steps in Proof............................... 213 19.8 Summary..................................... 216 20 The Theory of Doubly Periodic Functions 218 List of Figures 1 2.1 f(z) = z(z−1) ................................... 12 2.2 f(z) = Γ(z).................................... 12 2.3 f(z) = ζ(z).................................... 12 2.4 ez as a Mapping C ! C ............................. 20 2.5 Branches of Log(z)................................ 22 2.6 γ .......................................... 23 2.7 p(z − 1)z(z + 1)................................. 25 4.1 Various Integration Paths............................ 32 5.1 k-dimensional Sets................................ 38 5.2 k-dimensional Sets................................ 38 5.3 Defining f ..................................... 41 5.4 Defining f ..................................... 41 5.5 Defining f ..................................... 42 5.6 Proving Green's Theorem............................ 42 5.7 Region with Holes................................ 43 5.8 Extended Green Theorem............................ 43 6.1 Cauchy's Formula................................. 45 6.2 Goursat's Argument............................... 49 7.1 Homotopic Paths................................. 53 7.2 Star Shaped Regions............................... 53 7.3 Simply Connected Sets.............................. 54 dg 7.4 f = dz Locally.................................. 56 7.5 A Primitive for the Homotopy.......................... 59 8.1 A Complicated Curve.............................. 64 8.2 A Curve with Zero Integral............................ 65 8.3 A Region with Three Boundary Curves.................... 66 5 LIST OF FIGURES 6 8.4 Small Path Adjustment............................. 66 8.5 Main Diagram of the Proof........................... 67 8.6 A Limit...................................... 68 8.7 Another Limit.................................. 69 9.1 Existence of Laurent Series........................... 76 9.2 The Residue Theorem.............................. 81 9.3 Contour Integral One.............................. 82 9.4 Contour Integral Two.............................. 83 9.5 Proof of the Residue Theorem.......................... 83 10.1 Stereographic Projection............................. 86 10.2 Proof of Inverse Function Theorem....................... 89 df 10.3 Conformal Maps Near a Zero of dz ....................... 91 10.4 Uniform