Problems and Solutions for Complex Analysis / by Rami Shakarchi.

Format:

Book

Author/Creator:

Shakarchi, Rami, Author.

Contributor:

Lang, Serge, 1927-2005.

Language:

English

Subjects (All):

Mathematical analysis.

Analysis.

Local Subjects:

Analysis.

Physical Description:

1 online resource (XI, 246 p. 17 illus.)

Edition:

1st ed. 1999.

Place of Publication:

New York, NY : Springer New York : Imprint: Springer, 1999.

Language Note:

English

Summary:

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 .

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.

Notes:

"With 46 illustrations."

Related Work:

Lang, Serge, 1927-2005. Complex analysis.

ISBN:

1-4612-1534-X