Problems and Solutions for Complex Analysis

This book contains all the exercises and solutions of Serge Lang's Complex Analy­ sis. Chapters I through VITI of Lang's book contain the material of an introductory course at the undergraduate level and the reader will find exercises in all of the fol­ lowing topics: power series, Cauchy's theorem, Laurent series, singularities and meromorphic functions, the calculus of residues, conformal mappings and har­ monic functions. Chapters IX through XVI, which are suitable for a more advanced course at the graduate level, offer exercises in the following subjects: Schwarz re­ flection, analytic continuation, Jensen's formula, the Phragmen-LindelOf theorem, entire functions, Weierstrass products and meromorphic functions, the Gamma function and the Zeta function. This solutions manual offers a large number of worked out exercises of varying difficulty. I thank Serge Lang for teaching me complex analysis with so much enthusiasm and passion, and for giving me the opportunity to work on this answer book. Without his patience and help, this project would be far from complete. I thank my brother Karim for always being an infinite source of inspiration and wisdom. Finally, I want to thank Mark McKee for his help on some problems and Jennifer Baltzell for the many years of support, friendship and complicity. Rami Shakarchi Princeton, New Jersey 1999 Contents Preface vii I Complex Numbers and Functions 1 1. 1 Definition . . . . . . . . . . 1 1. 2 Polar Form . . . . . . . . . 3 1. 3 Complex Valued Functions . 8 1. 4 Limits and Compact Sets . . 9 1. 6 The Cauchy-Riemann Equations .

Table of Contents:
I Complex Numbers and Functions.- I.1 Definition.- I.2 Polar Form.- I.3 Complex Valued Functions.- I.4 Limits and Compact Sets.- I.6 The Cauchy-Riemann Equations.- II Power Series.- II.1 Formal Power Series.- II.2 Convergent Power Series.- II.3 Relations Between Formal and Convergent Series.- II.4 Analytic Functions.- II.5 Differentiation of Power Series.- II.6 The Inverse and Open Mapping Theorems.- III Cauchy’s Theorem, First Part.- III.1 Holomorphic Functions on Connected Sets.- III.2 Integrals over Paths.- III.5 The Homotopy Form of Cauchy’s Theorem.- III.6 Existence of Global Primitives Definition of the Logarithm.- III.7 The Local Cauchy Formula.- IV Winding Numbers and Cauchy’s Theorem.- IV.2 The Global Cauchy Theorem.- V Applications of Cauchy’s Integral Formula.- V.1 Uniform Limits of Analytic Functions.- V.2 Laurent Series.- V.3 Isolated Singularities.- VI Calculus of Residues.- VI.1 The Residue Formula.- VI.2 Evaluation of Definite Integrals.- VII Conformal Mappings.- VII.2 Analytic Automorphisms of the Disc.- VII.3 The Upper Half Plane.- VII.4 Other Examples.- VII.5 Fractional Linear Transformations.- VIII Harmonic Functions.- VIII.1 Definition.- VIII.2 Examples.- VIII.3 Basic Properties of Harmonic Functions.- VIII.4 The Poisson Formula.- VIII.5 Construction of Harmonic Functions.- IX Schwarz Reflection.- IX.2 Reflection Across Analytic Arcs.- X The Riemann Mapping Theorema.- X.1 Statement of the Theorem.- X.2 Compact Sets in Function Spaces.- XI Analytic Continuation along Curves.- XI.1 Continuation Along a Curve.- XI.2 The Dilogarithm.- XII Applications of the Maximum Modulus Principle and Jensen’s Formula.- XII.1 Jensen’s Formula.- XII.2 The Picard-Borel Theorem.- XII.6 The Phragmen-Lindelof and Hadamard Theorems.- XIII Entire and MeromorphicFunctions.- XIII.1 Infinite Products.- XIII.2 Weierstrass Products.- XIII.3 Functions of Finite Order.- XIII.4 Meromorphic Functions, Mittag-Leffler Theorem.- XV The Gamma and Zeta Functions.- XV.1 The Differentiation Lemma.- XV.2 The Gamma Function.- XV.3 The Lerch Formula.- XV.4 Zeta Functions.- XVI The Prime Number Theorem.- XVI.1 Basic Analytic Properties of the Zeta Function.- XVI.2 The Main Lemma and its Application.