Problems and Solutions for Complex Analysis

This volume contains all the exercises, and their solutions, for Serge Lang's fourth edition of "Complex Analysis," ISBN 0-387-98592-1. The problems in the first 8 chapters are suitable for an introductory course at the undergraduate level and cover the following topics: power series, Cauchy's theorem, Laurent series, singularities and meromorphic functions, the calculus of residues, conformal mappings, and harmonic functions. The material in Chapters 9-16 is more advanced. The reader will find problems on Schwartz reflection, analytic continuation, Jensen's formula, the Phragmen-Lindeloef theorem, entire functions, Weierstrass products and meromorphic functions, the Gamma function and Zeta function. This volume also serves as an independent source of problems with detailed answers beneficial for anyone interested in learning 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. The present book is meant as a text for a course on complex analysis at the advanced undergraduate level, or first-year graduate level. The first half, more or less, can be used for a one-semester course addressed to undergraduates. The second half can be used for a second semester, at either level. Somewhat more material has been included than can be covered at leisure in one or two terms, to give opportunities for the instructor to exercise individual taste, and to lead the course in whatever directions strikes the instructor's fancy at the time as well as extra read­ ing material for students on their own. A large number of routine exer­ cises are included for the more standard portions, and a few harder exercises of striking theoretical interest are also included, but may be omitted in courses addressed to less advanced students. In some sense, I think the classical German prewar texts were the best (Hurwitz-Courant, Knopp, Bieberbach, etc.) and I would recommend to anyone to look through them. More recent texts have emphasized connections with real analysis, which is important, but at the cost of exhibiting succinctly and clearly what is peculiar about complex analysis: the power series expansion, the uniqueness of analytic continuation, and the calculus of residues. Pt. 1. Basic Theory -- Ch. I. Complex Numbers And Functions -- Ch. Ii. Power Series -- Ch. Iii. Cauchy's Theorem, First Part -- Ch. Iv. Winding Numbers And Cauchy's Theorem -- Ch. V. Applications Of Cauchy's Integral Formula -- Ch. Vi. Calculus Of Residues -- Ch. Vii. Conformal Mappings -- Ch. Viii. Harmonic Functions -- Pt. 2. Geometric Function Theory -- Ch. Ix. Schwarz Reflection -- Ch. X. The Riemann Mapping Theorem -- Ch. Xi. Analytic Continuation Along Curves -- Pt. 3. Various Analytic Topics -- Ch. Xii. Applications Of The Maximum Modulus Principle And Jensen's Formula -- Ch. Xiii. Entire And Meromorphic Functions -- Ch. Xiv. Elliptic Functions -- Ch. Xv. The Gamma And Zeta Functions -- Ch. Xvi. The Prime Number Theorem -- App. 1. Summation By Parts And Non-absolute Convergence -- App. 2. Difference Equations -- App. 3. Analytic Differential Equations -- App. 4. Fixed Points Of A Fractional Linear Transformation -- App. 5. Cauchy's Formula For C[superscript Infinity] Functions -- App. 6. Cauchy's Theorem For Locally Integrable Vector Fields. Serge Lang. Includes Bibliographical References (p. [478]) And Index. All the exercises plus their solutions for Serge Lang's fourth edition of "Complex Analysis," ISBN 0-387-98592-1. The problems in the first 8 chapters are suitable for an introductory course at undergraduate level and cover power series, Cauchy's theorem, Laurent series, singularities and meromorphic functions, the calculus of residues, conformal mappings, and harmonic functions. The material in the remaining 8 chapters is more advanced, with problems on Schwartz reflection, analytic continuation, Jensen's formula, the Phragmen-Lindeloef theorem, entire functions, Weierstrass products and meromorphic functions, the Gamma function and Zeta function. Also beneficial for anyone interested in learning complex analysis. "This is the fourth edition of Serge Lang's Complex Analysis. The first part of the book covers the basic material of complex analysis, and the second covers many special topics, such as the Riemann Mapping Theorem, the gamma function, and analytic continuation. Power series methods are used more systematically than in other texts, and the proofs using these methods often shed more light on the results than the standard proofs do. The first part of Complex Analysis is suitable for an introductory course on the undergraduate level, and the additional topics covered in the second part give the instructor of a graduate course a great deal of flexibility in structuring a more advanced course."--BOOK JACKET Now in its fourth edition, the first part of this book is devoted to the basic material of complex analysis, while the second covers many special topics, such as the Riemann Mapping Theorem, the gamma function, and analytic continuation. Power series methods are used more systematically than is found in other texts, and the resulting proofs often shed more light on the results than the standard proofs. While the first part is suitable for an introductory course at undergraduate level, the additional topics covered in the second part give the instructor of a gradute course a great deal of flexibility in structuring a more advanced course. This book is an independent source of problems with detailed answers beneficial for anyone interested in learning complex analysis. The problems cover such topics as: power series, Cauchy's theorem, Laurent series, singularities and meromorphic functions, the calculus of residues, conformal mappings, harmonic functions, and Jensen's formula. All the exercises from this volume were derived from Serge Lang's fourth edition of Complex Analysis and can also serve as a companion guide to it. The complex numbers are a set of objects which can be added and multiplied, the sum and product of two complex numbers being also a complex number, and satisfy the following conditions. This well-established book covers the basic material of complex analysis, plus many special topics, such as the Riemann mapping theorem, the gamma function, and analytic continuation.