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Lecture Notes in Complex Analysis

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Lecture Notes in Complex Analysis Lecture Notes in Complex Analysis Based on lectures by Dr Sheng-Chi Liu Throughout these notes, signifies end proof, N signifies end of exam- ple, and marks the end of exercise. The exposition in these notes is based in part on [Con78]. This is also where most exercises are from, and is a good source of further exercises for the intrepid reader. Table of Contents Table of Contents i Lecture 1 Review of Basic Complex Analysis 1 1.1 Basics . 1 1.2 Analytic functions . 2 1.3 Cauchy–Riemann equations . 3 Lecture 2 Möbius Transformations 5 2.1 Conformal mappings . 5 2.2 Möbius transformations . 6 Lecture 3 Power Series 10 3.1 Möbius transformations preserve lines and circles . 10 3.2 Power series representation of analytic functions . 11 Lecture 4 Entire Functions 13 4.1 Properties of analytic and entire functions . 13 Lecture 5 Cauchy’s Integral Theorem 18 5.1 Winding numbers . 18 5.2 Homotopy . 21 Lecture 6 Counting Zeros 21 6.1 Simply connected sets and Simple closed curve . 21 6.2 Counting zeros . 23 Lecture 7 Singularities 26 7.1 Classifying isolated singularities . 26 Lecture 8 Laurent Expansion 29 8.1 Laurent expansion around isolated singularity . 29 8.2 Residues . 33 Notes by Jakob Streipel. Last updated November 11, 2020. i ii TABLE OF CONTENTS Lecture 9 The Argument Principle 34 9.1 Residue calculus . 34 9.2 The argument principle . 35 Lecture 10 Bounds of Analytic Functions 38 10.1 Simple bounds . 38 10.2 Automorphisms of the unit disk . 40 10.3 Automorphisms of the upper half-plane . 42 Lecture 11 Hadamard Three-Lines Theorem 44 11.1 Generalising the maximum modulus principle . 44 Lecture 12 Phragmén–Lindelöf Principle 46 12.1 Further generalising the maximum modulus principle . 46 Lecture 13 The Space of Analytic Functions 50 13.1 The topology of C(G, C) ....................... 50 13.2 The space of analytic functions . 52 Lecture 14 Compactness in H(G) 52 14.1 Compactness in the space of analytic functions . 52 14.2 Montel’s theorem . 54 Lecture 15 The Space of Meromorphic Functions 56 15.1 The topology of C(G, C∞) ...................... 56 Lecture 16 Compactness in M(G) 58 16.1 Compactness in the space of meromorphic functions . 58 Lecture 17 Compactness in M(G), continued 62 17.1 Compactness in the space of meromorphic functions, finalised . 62 17.2 Riemann mapping theorem . 64 Lecture 18 The Riemann Mapping Theorem 64 18.1 Proving the Riemann mapping theorem . 64 18.2 Entire functions . 68 Lecture 19 Infinite Products 69 19.1 Review of infinite products . 69 Lecture 20 Weierstrass Factorisation Theorem 74 20.1 Elementary factors . 74 Lecture 21 Rank and Genus 79 21.1 Jensen’s formula . 79 21.2 The genus and rank of entire functions . 81 Lecture 22 Order 84 22.1 The order of entire functions . 84 22.2 Hadamard’s factorisation theorem . 85 Lecture 23 Range of Analytic Functions 88 TABLE OF CONTENTS iii 23.1 Proof of Hadamard’s factorisation theorem . 88 23.2 The range of entire functions . 89 23.3 The range of an analytic function . 90 Lecture 24 Bloch’s and Landau’s Constants 92 24.1 Proof of Bloch’s theorem . 92 24.2 The Little Picard theorem . 96 Lecture 25 Toward the Little Picard Theorem 97 25.1 Another lemma . 97 Lecture 26 The Little Picard Theorem 98 26.1 Proof of the Little Picard theorem . 98 26.2 Schottky’s theorem . 99 Lecture 27 The Great Picard Theorem 101 27.1 The range of entire functions, revisited . 101 27.2 Runge’s theorem . 104 Lecture 28 Mittag-Leffler’s Theorem 105 28.1 Runge’s theorem . 105 28.2 Mittag-Leffler’s theorem . 108 Lecture 29 Analytic Continuation 110 29.1 Analytic continuation . 110 29.2 Dirichlet series . 112 Lecture 30 Perron’s Formula 114 30.1 Uniqueness of Dirichlet series . 114 30.2 Perron’s formula . 116 References 119 Index 120 iv TABLE OF CONTENTS REVIEW OF BASIC COMPLEX ANALYSIS 1 Lecture 1 Review of Basic Complex Analysis 1.1 Basics Throughout√ we will denote complex numbers by z = x + iy, where x, y ∈ R and i = −1. We call x = Re(z) and y = Im(y), the real and imaginary parts of z, respectively. Moreover we call z = x − iy the complex conjugate of z and |z| = px2 + y2 = |z| the absolute value or modulus of z. Note, which occasionally comes in handy, that |z|2 = zz. Also commonplace is to discuss polar representation of complex numbers. To this end, let θ ∈ R and define eiθ := cos(θ) + i sin(θ). This way eiθ is a point on the unit circle in the complex plane, and so naturally we can scale this by a real number to reach any point in the plane. Since the complex plane C has an absolute value |·|, as per above, we also have an induced metric, namely for z1, z2 ∈ C we define the distance d(z1, z2) = |z1 − z2|. Under this metric (C, d) is a complete metric space. We will take a moment to discuss some commonly useful functions on the complex plane. The first suspect we have almost seen above already, namely ez. If z = x + iy, then of course ez = ex+iy = ex · eiy = ex(cos(y) + i sin(y)), so |ez| = ex = eRe(z). Close cousin of the exponential function are the basic trigonometric func- tions; we would like for sin(z) and cos(z) to satisfy ez = cos(z) + i sin(z) (and, correspondingly, eiz = cos(z) − i sin(z)), and so to that end we define eiz + e−iz cos(z) := 2 and eiz − e−iz sin(z) := . 2i Another close cousin to the exponential function is the natural logarithm. This, for the first time in our brief exploration, is where things get a little bit hairy. We of course would like for elog(z) = z, since we want log to be the inverse function of the exponential. Now, writing z = |z|eiθ, where θ = arg(z), its argument, we can try to puzzle out what log(z) should look like, by setting log(z) = u + iv and studying elog(z). Since we then have elog(z) = eu+iv = eu · eiv, and in turn we want elog(z) = z = |z|eiθ we have, comparing sides, eu = |z|, i.e., u = log|z|, and eiv = eiθ. This last one is where it gets hairy: this does not imply v = θ, but more loosely that v = θ + k2π for any k ∈ Z. Thus log(z) := log|z| + i(θ + k2π) Date: August 20th, 2019. 2 REVIEW OF BASIC COMPLEX ANALYSIS for k ∈ Z, meaning this right-hand side is not unique, so log(z) is not well- defined, and hence not a function. To resolve this we need to choose a branch of log(z), by which we mean deciding in what range we let the argument live; so long as it’s any open interval of length 2π we are fine. For instance, between 0 and 2π would be fine, as would −π to π. 1.2 Analytic functions Definition 1.2.1 (Complex differentiable). Let G ⊂ C be an open set. A function f : G → C is differentiable at z ∈ G if f(z + h) − f(z) f 0(z) := lim h→0 h exists. Note that, crucially, since h is a complex number, this limit must exist along all paths. Definition 1.2.2 (Analytic function). Let G ⊂ C be an open set. A function f : G → C is analytic if it is continuously differentiable in G, i.e., f 0(z) exists and is continuous for all z ∈ G. Remark 1.2.3. (i) Every (complex) differentiable function is analytic (so the requirement of continuously above is not necessary). This, of course, is not true in real analysis. Take, for example, ( x2 sin( 1 ), if x 6= 0, f(x) = x 0, if x = 0. Then f 0(x) exists everywhere, but is not continuous at x = 0. (ii) Every analytic function is infinitely differentiable, and has a power series representation. This, again, is not true in the real case. Take, say, ( 0, if x ≤ 0, f(x) = − 1 e x2 , if x > 0. Then f (n)(0) = 0 for all n ∈ N, meaning that the Taylor series centred at x = 0 is identically zero, but f is not identically zero near 0, so f(x) has no power series there (or, another way of putting it, the radius of convergence of its power series around x = 0 is zero). 2 Exercise 1.1. Show that f(z) = |z| has a derivative only at z = 0. ∞ P n Proposition 1.2.4. Suppose that an(z −a) has radius of convergence R > n=0 ∞ P n 0. Then f(z) := an(z − a) is analytic in |z − a| < R, and moreover n=0 ∞ (k) X n−k f (z) = n(n − 1) ... (n − k + 1)an(z − a) n=0 for |z − a| < R (meaning that the derivative can be taken termwise), and in addition f (n)(a) a = . n n! 1.3 Cauchy–Riemann equations 3 Example 1.2.5. As expected we have ∞ X zn ez = n! n=0 for all z ∈ C, i.e., R = ∞. Also as expected, ∞ ∞ X nzn X zn (ez)0 = = = ez n! n! n=1 n=0 by reindexing the first sum. Hence this is analytic. N Example 1.2.6. Similarly, we then get eiz − e−iz z3 z5 z2n−1 sin(z) = = z − + − · · · + (−1)n + ..
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