Functions of a Complex Variable II Math 562, Spring 2020 Jens Lorenz April 27, 2020 Department of Mathematics and Statistics, UNM, Albuquerque, NM 87131 Contents 1 Notations 5 2 The Schwarz Lemma and Aut(D) 6 2.1 Schwarz Lemma . .6 2.2 Biholomorphic Maps and Automorphisms . .7 2.3 The Automorphism Group of D ...................7 2.4 The Schwarz{Pick Lemma . .9 2.5 Remarks . 11 3 Linear Fractional Transformations and the Riemann Sphere 12 3.1 The Riemann Sphere . 12 3.2 Linear Fractional Transformations . 14 3.3 The Automorphisms of the Riemann Sphere . 16 3.4 A Remarkable Geometric Property of Linear Fractional Trans- formations . 17 3.4.1 Analytical Description of Circles . 18 3.4.2 Circles under Reciprocation . 18 3.4.3 Straight Lines under Reciprocation . 20 3.5 Example: The Cayley Transform in the Complex Plane . 21 3.6 The Cayley Transform of a Matrix . 22 4 The Riemann Mapping Theorem 24 4.1 Statement and Overview of Proof . 24 4.2 The Theorem of Arzela{Ascoli . 26 4.3 Montel's Theorem . 29 4.4 Auxiliary Results on Logarithms and Square Roots . 32 4.5 Construction of a Map in ..................... 33 4.6 A Bound of f 0(P ) for all Ff .................. 35 4.7 Application ofj thej Theorems2 of F Montel and Hurwitz . 36 4.8 Proof That f is Onto . 37 1 4.9 Examples of Biholomorphic Mappings . 39 5 An Introduction to Schwarz–Christoffel Formulas 41 5.1 General Power Functions . 41 5.2 An Example of a Schwarz–Christoffel Map . 42 6 Meromorphic Functions with Prescribed Poles 47 6.1 Meromorphic Functions on C .................... 47 6.2 The Mittag–Leffler Theorem . 47 6.3 An Example . 50 7 Infinite Products 52 7.1 Infinite Products of Complex Numbers . 52 7.1.1 Two Definitions of Convergence . 52 7.1.2 Examples . 53 7.2 Infinite Products of Numbers: Convergence Theory . 56 7.3 Infinite Products of Functions . 61 7.4 Example: The Product Formula for the Sine Function . 64 7.5 Euler's Constant . 65 7.6 The Gauss Formula for Γ(z) and Weierstrass' Product Formula for 1=Γ(z)............................... 66 7.6.1 A Formula for (Γ0=Γ)0 .................... 70 7.7 Entire Functions with Prescribed Zeros . 71 7.7.1 Construction of f(z); Motivation . 72 7.7.2 Weierstrass' Canonical Factors . 72 8 The Bernoulli Numbers and Applications 77 8.1 The Bernoulli Numbers . 77 8.2 The Taylor Series for z cot z in Terms of Bernoulli Numbers . 79 8.3 The Mittag–Leffler Expansion of πz cot(πz)............ 80 8.4 The Values of ζ(2m)......................... 81 8.5 Sums of Powers and Bernoulli Numbers . 81 9 The Riemann Zeta{Function 84 9.1 Definition for Re s > 1........................ 84 9.2 Simple Bounds of ζ(s) for s > 1................... 84 9.3 Meromorphic Continuation of ζ(z) to Re z > 0 .......... 85 9.4 Analytic Continuation of ζ(z) to C 1 .............. 87 9.5 Euler's Product Formula for ζ(z)..................n f g 90 P 1 9.6 The Sum p and the Prime Number Theorem . 93 9.7 Auxiliary Results about Fourier Transformation: Poisson's Sum- mation Formula . 94 9.8 A Functional Equation . 96 9.9 Growth Estimates for the ζ{Function . 98 2 10 Analytic Continuation 101 10.1 Analytic Continuation Using the Cauchy Riemann Equations . 101 10.2 Exponential Decay of Fourier Coefficients and the Strip of Ana- lyticity . 101 10.3 The Schwarz Reflection Principle . 101 10.4 Examples for Analytic Continuation . 101 10.5 Riemann Surfaces: Intuitive Approach . 101 10.6 Riemann Surfaces: Germs, Sheafs, and Fibers . 101 11 Fourier Series 102 11.1 Convergence Results: Overview . 102 11.2 The Dirichlet Kernel . 104 11.3 The Fej´erKernel . 105 11.4 Convergence of σnf in Maximum Norm . 107 11.5 Weierstrass' Approximation Theorem for Trigonometric Polyno- mials . 108 11.6 Convergence of Snf in L2 ...................... 108 11.7 The Lebesgue Constant of the Dirichlet Kernel . 110 11.7.1 The Lebesgue Constant as Norm of a Functional . 112 11.8 Divergence of the Fourier Series of a Continuous Function at a Point.................................. 112 11.9 Isomorphy of L2(0; 1) and l2 ..................... 114 11.10Convergence of Snf in Maximum Norm . 116 11.11Smoothness of f(x) and Decay of f^(k)............... 117 11.12Exponential Decay of f^(k) and Analyticity of f .......... 119 11.13Divergence of Snf(0): Explicit Construction of f ......... 121 11.14Fourier Series and the Dirichlet Problem for Laplace's Equation on the Unit Disk . 123 12 Fourier Transformation 124 12.1 Motivation: Application to PDEs . 124 12.2 The Fourier Transform of an L1 Function . 127 12.3 The Riemann{Lebesgue Lemma . 128 12.4 The Fourier Transform on the Schwartz Space ......... 130 12.5 The Fourier Inversion Formula on the SchwartzS Space: Prepara- tions . 131 12.6 The Fourier Inversion Formula on the Schwartz Space . 134 12.7 Operators . 135 12.8 Elementary Theory of Tempered Distributions . 136 12.8.1 Ordinary Functions as Tempered Distributions . 137 12.9 The Fourier Transform on 0: An Example Using Complex Vari- ables . .S . 137 12.10Decay of the Fourier Transform of f and Analyticity of f .... 138 12.11The Paley{Wiener Theorem . 138 12.12The Laplace Transform and Its Inversion . 139 3 13 Growth and Zeros of Entire Functions 140 13.1 Jensen's Formula . 140 13.2 The Order of Growth . 142 13.3 Zeros and Growth Estimates . 143 13.4 Hadamard's Factorization Theorem . 145 13.5 Entire Functions of Non{Integer Order of Growth . 148 13.6 Proof of Hadamard's Factorization Theorem . 148 14 The Prime Number Theorem 155 14.1 Functions . 155 14.2 Reduction to Asymptotics of 1(x)................. 157 14.3 Integral Representation of 1(x).................. 161 14.4 Auxiliary Results . 161 14.5 The ζ{Function has no Zero on the Line s = 1 + it ........ 163 15 Complex Variables and PDEs 166 15.1 2D Irrotational Euler Flows . 166 15.2 Laplace Equation . 171 15.2.1 Derivation of the Poisson Kernel via Complex Variables . 171 15.2.2 Derivation of the Poisson Kernel via Separation of Variables175 16 Rearrangement of Series 176 16.1 Rearrangement of Absolutely Convergent Series . 176 16.2 Interchanging Double Sums . 177 16.3 Rearrangement of a Double Series . 180 4 1 Notations R field of real numbers C field of complex numbers D(z0;R) z : z z0 < R : open disk of radius R centered at z0 ¯ f j − j g D(z0;R) z : z z0 R : closed disk of radius R centered at z0 @D(z ;R) fz : jz − z j= ≤ Rg : boundary of disk of radius R centered at z 0 f j − 0j g 0 D D = D(0; 1) = z : z < 1 : open unit disk f j j g H H = z = x + iy : y > 0 : open upper half{plane f g H(U) set of all holomorphic functions f : U C where U C is open ! ⊂ C(U) set of all continuous functions f : U C where U C is any set R x ds ! ⊂ ln(x) for real positive x: ln(x) = 1 s log(z) complex logarithm 5 2 The Schwarz Lemma and Aut(D) 2.1 Schwarz Lemma Let D = D(0; 1) denote the open unit disk. In the Schwarz Lemma one considers functions f H(D) with f(0) = 0 and f(D) D¯. The Schwarz Lemma then says that the2 estimate f(z) 1 can be sharpened.⊂ We recall the maximumj j modulus ≤ theorem, which will be used in the proof of the Schwarz Lemma. Theorem 2.1 (Maximum Modulus Theorem) Let U C denote an open, con- nected set and let g H(U). If there exists a point z⊂ U with 2 0 2 g(z) g(z ) for all z U j j ≤ j 0 j 2 then g(z) is constant in U. In other words, only constant holomorphic functions attain their maximal value. This follows from the open mapping theorem: A holomorphic function g(z) on an open, connected set U maps open subsets of U to open sets, unless g(z) is constant. Theorem 2.2 (Schwarz Lemma) Let f H(D) satisfy 2 a) f(z) 1 for all z D; b) fj (0) =j ≤ 0. 2 0 Then f(z) z for all z D and f (0) 1. In addition, if j j ≤ j j 2 j j ≤ f(z ) = z j 0 j j 0j 0 for some z0 D 0 or if f (0) = 1, then f is a rotation, i.e., 2 n f g j j f(z) = αz for some α C with α = 1. 2 j j Proof: Set f(z)=z; 0 < z < 1 g(z) = f 0(0); z =j 0 j Since f(0) = 0 we have 1 X f(z) = a zj for z < 1 j j j j=1 and obtain that g H(D). Let 0 < " < 1 and consider g in the closed disk 2 D¯(0; 1 ") D. Since f(z) 1 in D one obtains that − ⊂ j j ≤ 1 g(z) for z = 1 ": j j ≤ 1 " j j − − By the maximum modulus theorem we conclude that 6 1 g(z) for z 1 ": j j ≤ 1 " j j ≤ − − As " 0 this yields: ! g(z) 1 for z < 1 : j j ≤ j j Therefore, f(z) z for z < 1 and f 0(0) 1 : j j ≤ j j j j j j ≤ Now assume that f(z0) = z0 for some z0 D; z0 = 0; or assume that f 0(0) = 1.