Lectures on complex analysis Alexander Kupers April 30, 2020 Abstract These are the collected lecture notes for Math 113. i Contents 1 Complex numbers and holomorphic functions1 1.1 Complex numbers ............................... 1 1.1.1 The algebra of complex numbers................... 2 1.1.2 The geometry of complex numbers.................. 4 1.2 Holomorphic functions............................. 6 1.2.1 Examples of holomorphic functions.................. 7 1.2.2 Integration of holomorphic functions................. 8 1.3 The Riemann zeta-function .......................... 9 2 Functions on the complex plane 11 2.1 The complex plane............................... 11 2.1.1 Recollection: the algebra of complex numbers............ 11 2.1.2 The topology of complex numbers .................. 11 2.2 Holomorphic functions............................. 14 2.2.1 Continuity ............................... 14 2.2.2 Complex differentiation ........................ 15 2.2.3 The Cauchy–Riemann equations ................... 16 3 Integration and Goursat’s theorem 18 3.1 Integration along curves............................ 18 3.1.1 Examples of integrals ......................... 20 3.2 Goursat’s theorem ............................... 23 3.2.1 Goursat’s theorem for triangles.................... 23 3.2.2 A down-to-earth argument in support of Goursat’s theorem . 25 4 Cauchy’s integral formula 27 4.1 Holomorphic functions have primitives on disks............... 27 4.2 Computing some integrals........................... 30 4.3 Cauchy integral formula............................ 32 5 Applications of Cauchy’s integral formula 35 5.1 Cauchy integral formula............................ 35 ii Contents iii 5.1.1 Liouville’s theorem and the fundamental theorem of algebra . 35 5.2 Holomorphic functions are analytic...................... 36 5.2.1 Power series............................... 36 5.2.2 Analytic functions ........................... 38 5.2.3 Analytic continuation ......................... 40 6 Morera’s theorem and its applications 42 6.1 Morera’s theorem................................ 42 6.2 Constructing holomorphic functions by limits................ 42 6.2.1 Sequences................................ 42 6.2.2 Sums .................................. 44 6.2.3 Integrals................................. 45 6.2.4 Pointwise convergence......................... 45 6.3 Schwarz reflection ............................... 46 7 Poles 49 7.1 Poles and zeroes ................................ 49 7.1.1 Isolated singularities.......................... 49 7.1.2 Zeroes and poles ............................ 49 7.1.3 Laurent series.............................. 51 7.2 The residue formula .............................. 52 7.3 Integrals using the residue formula...................... 53 7.3.1 Zeroes and poles ............................ 55 7.4 The residue formula .............................. 56 8 Singularities 58 8.1 Removable singularities ............................ 58 8.2 Essential singularities ............................. 59 8.2.1 Application: the automorphisms of C . 60 8.3 Meromorphic functions and the complex projective plane ......... 60 1 8.3.1 Application: the automorphisms of CP . 62 9 The argument principle 64 9.1 The argument principle ............................ 64 9.2 Rouch´e’stheorem................................ 65 9.3 The open mapping theorem.......................... 66 9.4 The maximum modulus principle....................... 66 10 Homotopy invariance and the logarithm 68 10.1 Integrals along paths are homotopy-invariant ................ 68 10.2 The logarithm.................................. 70 10.2.1 Logarithms of holomorphic functions................. 72 11 Pre-midterm recap 73 11.1 Holomorphic functions and their properties ................. 73 11.2 Constructing holomorphic functions ..................... 74 iv Contents 11.3 The classification of singularities ....................... 74 11.4 Evaluation of integrals............................. 75 11.5 Integral formula’s and growth behavior of holomorphic functions . 76 12 Growth of zeroes 77 12.1 The mean value property ........................... 77 12.2 Jensen’s formula ................................ 78 12.3 Functions of finite order............................ 80 12.4 An application of Jensen’s formula to polynomials ............. 82 13 Weierstrass’ construction 84 13.1 Infinite products ................................ 84 13.1.1 The product formula for the sine................... 85 13.2 Weierstrass’ construction............................ 87 14 Hadamard factorization theorem 89 14.1 Hadamard factorization theorem ....................... 89 14.1.1 Preparatory lemmas.......................... 90 14.1.2 The proof................................ 92 14.2 Picard’s little theorem for functions of finite order ............. 93 15 The Gamma function 94 15.1 The Gamma function ............................. 94 15.1.1 Analytic extension........................... 94 15.1.2 Symmetry................................ 96 15.1.3 Growth ................................. 96 15.1.4 Product formula ............................ 98 15.2 Mellin transforms................................ 98 16 The Riemann zeta function 100 16.1 The Riemann zeta function ..........................100 16.1.1 The theta function...........................100 16.1.2 Expressing ζ in terms of θ . 102 16.1.3 Estimates on the line Re(s) = 1....................104 17 The zeroes of the Riemann zeta function 106 17.1 Recollection on the prime number theorem . 106 17.1.1 The Riemann zeta function as a product . 106 17.2 The trivial zeroes of Riemann zeta function . 108 17.3 The Riemann zeta function has no zeroes on Re(s) = 1 . 108 17.3.1 Some additional estimates.......................110 18 Tchebychev’s functions 111 18.1 Tchebychev’s functions.............................111 18.2 Relating Tchebychev’s functions to ζ . 113 18.3 Proving the asymptotics of ψ1 . 115 Contents v 19 The proof of the prime number theorem 116 19.1 The asymptotics of ψ1(x) ...........................116 19.2 Zeroes of the Riemann zeta function and error terms for the prime number theorem.....................................118 20 Conformal equivalences 121 20.1 Conformal equivalences ............................121 20.2 Examples ....................................123 20.2.1 The half-plane is conformally equivalent to the unit disk . 123 20.2.2 Further examples............................124 20.3 The Riemann mapping theorem........................125 21 The Schwartz lemma and conformal automorphisms 126 21.1 Schwartz lemma ................................126 21.2 Conformal automorphisms of the upper half-plane and unit disk . 127 21.2.1 Conformal automorphisms.......................127 21.2.2 The conformal automorphisms of D . 127 21.2.3 The conformal automorphisms of H . 129 22 Several complex variables 133 22.1 Holomorphic functions of several complex variables . 133 22.2 The Hartogs extension theorem........................134 22.3 Domains of holomorphy and convergence of power series . 135 23 Final recap 138 23.1 Midterm material................................138 23.2 Zeroes of entire functions ...........................139 23.2.1 Jensen’s formula ............................139 23.2.2 Functions of finite order........................140 23.3 The Gamma function, Riemann zeta function, and the prime number theorem.....................................140 23.4 Conformal equivalences ............................141 Bibliography 142 Chapter 1 Complex numbers and holomorphic functions In this first chapter I will give you a taste of complex analysis, and recall some basic facts about the complex numbers. We define holomorphic functions, the subject of this course. These functions turn out to be much more well-behaved than the functions you have encountered in real analysis. We will mention the most striking such properties that we will prove during the courses. We also give many examples of holomorphic functions, ending with the most exciting example: the Riemann zeta-function. We will go over this material in more depth in the coming lectures. However, you can take a look at [SS03, Section 1.1], [MH87, Section 1.1 and 1.2] for more details about the complex numbers. 1.1 Complex numbers Complex numbers have their origin in attempts to understand the roots of polynomials. By the sixteenth century, mathematicians had found formulas for the solutions of the equation p(x) = 0, when p is a quadratic or cubic polynomial. For example, when p(x) = ax2 + bx + c such solutions x0 are given by the well-known quadratic formula √ −b ± b2 − 4ac x = . 0 2a When b2 − 4ac ≥ 0 the graph of p intersects the x-axis, and the above formula can be 2 interpreted as giving the x-coordinate of these intersections. When b − 4ac√ < 0 there are no such intersections, which is reflected by the fact that at first sight b2 − 4ac is meaningless when the expression b2 − 4ac, called the discriminant is negative. This suggests one way to interpret the quadratic formula: if the discriminant is positive it gives two roots, if the discriminant is zero it gives a single root, and if it is negative it gives no roots. However, it was observed that if one introduced a new number i satisfying i2 = −1 to the number system, and let square roots of negative numbers be multiples of i, they not only appeared to give roots for the polynomial p when the discriminant is negative, 1 2 Chapter 1 Complex numbers and holomorphic functions but could be fruitfully manipulated to solve various other algebraic