Problems and Solutions for Complex Analysis

ISBN-13: 9780387988313 / Angielski / Miękka / 1999 / 246 str.

Rami Shakarchi; R. Shakarchi

Opis

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.

Spis treści

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 Meromorphic Functions.- 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.