Solutions Complex Analysis

In the pages that follow there are: A. A revised and enlarged version of Problems in analysis (PIA) . (All typographical, stylistic, and mathematical errors in PIA and known to the writer have been corrected.) B. A new section COMPLEX ANALYSIS containing problems distributed among many of the principal topics in the theory of functions of a complex variable. C. A total of 878 problems and their solutions.

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 .

This is a collection of exercises in the theory of analytic functions, with completed and detailed solutions. We wish to introduce the student to applications and aspects of the theory of analytic functions not always touched upon in a first course. Using appropriate exercises show the students some aspects of what lies beyond a first course in complex variables. We also discuss topics of interest for electrical engineering students (for instance, the realization of rational functions and its connections to the theory of linear systems and state space representations of such systems). Examples of important Hilbert spaces of analytic functions (in particular the Hardy space and the Fock space) are given. The book also includes a part where relevant facts from topology, functional analysis and Lebesgue integration are reviewed.

A companion volume to the text "Complex Variables: An Introduction" by the same authors, this book further develops the theory, continuing to emphasize the role that the Cauchy-Riemann equation plays in modern complex analysis. Topics considered include: Boundary values of holomorphic functions in the sense of distributions; interpolation problems and ideal theory in algebras of entire functions with growth conditions; exponential polynomials; the G transform and the unifying role it plays in complex analysis and transcendental number theory; summation methods; and the theorem of L. Schwarz concerning the solutions of a homogeneous convolution equation on the real line and its applications in harmonic function theory.

The first part of the volume contains a comprehensive description of the theory of entire and meromorphic functions of one complex variable and its applications. It includes the fundamental notions, methods and results on the growth of entire functions and the distribution of their zeros, the Rolf Nevanlinna theory of distribution of values of meromorphic functions including the inverse problem, the theory of completely regular growth, the concept of limit sets for entire and subharmonic functions.

This book is devoted to classical and modern achievements in complex analysis. In order to benefit most from it, a first-year university background is sufficient; all other statements and proofs are provided.

This book highlights the mathematical and physical properties of acoustical sources with singularities located in the complex plane and presents the application of such special elements to solve acoustical radiation and scattering problems.