Modern Analysis Series Edited by Chung-Chun Yang AN INTRODUCTION TO COMPLEX ANALYSIS

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1 Modern Analysis Series Edited by Chung-Chun Yang AN INTRODUCTION TO COMPLEX ANALYSIS Classical and Modern Approaches Wolfgang Tutschke Harkrishan L. Vasudeva ««CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C.

2 Contents 1 Preliminaries The field of complex numbers Introduction Complex numbers as pairs of real numbers Solvability of z 2 = Definition of complex numbers using equivalence classes of polynomials The complex plane Geometrical Interpretation of complex numbers Absolute value. Conjugate complex numbers Interpretation of complex numbers as vectors Trigonometrie form of complex numbers Geometrical Interpretation of the produet of complex numbers Powers and roots of complex numbers Some special sets in the complex plane Examples Metrie Spaces The coneept of a metric space Open sets Convergence Closed subsets Definition of some more terms Completeness Compact sets Coverings Examples Mappings and funetions. Continuity 28 vii

3 viii Basic definitions The extended complex plane. Spherical distance Limits of functions Continuous functions Curves Connectivity Sequences of functions Infinite series Power series The complex exponential and trigonometric functions Logarithm and its Riemann Surface Remark on the concept of a topological space Examples Exercises to Chapter The classical approach Ordinary complex differentiation Complex differentiability Rules of differentiation The Cauchy-Riemann System Holomorphic functions Differentiation of Power Series Differentiation of inverse functions Preliminaries of the Integral Calculus Line integrals of real valued functions The Green-Gauss Integral Theorem Line integrals of complex valued functions Complex Integral Theorems Cauchy's Integral Theorem Cauchy's Integral Formula Exercises to Chapter 2 116

4 IX An alternative approach Partial complex differentiations Linearization of functions of one real variable Linearization of functions depending on several real variables Linearization of functions depending on one complex variable Definition of partial complex derivatives Differentiability rules for partial complex derivatives Complex Green-Gauss Integral Theorems * Generalized Cauchy Integral Formula * Application of complex version of the Green-Gauss Formula to functions having isolated singularities The limit of the domain integral. Schmidt's Inequality The limit of the line integral An integral representation formula for continuously differentiable functions The classical Cauchy Integral Formula * Another approach to Cauchy's Integral Formula A second proof of Cauchy's Integral Theorem Comparison * Partial complex derivatives of functions having an ordinary complex derivative Ordinary complex differentiability of Solutions of the Cauchy-Riemann System Exercises to Chapter Local properties Existence of higher order derivatives A method for proving local properties of holomorphic functions The holomorphy of line integrals with respect to complex parameters Cauchy's Integral Formula for the derivatives of a holomorphic function Local power series representation Power series representation of the Cauchy kernel

5 X Local power series for holomorphic functions and an integral representation of the coefficients Cauchy's estimate of the coefficients The power series of the product of two holomorphic functions Division of power series Distribution of zeros The Weierstrass Convergence Theorem Statement of the problem Formulation and proof Termwise differentiability Connexion with plane Potential Theory Holomorphic functions as Solutions of the Laplace equation Representation of the Laplace Operator by partial complex differentiations Complex Integral Theorems revisited * Goursat's Theorem G. Fichera's proof of the Goursat Theorem A measure-theoretic approach to Cauchy's Integral Theorem Consequences of Complex Integral Theorems under weaker assumptions Exercises to Chapter Global properties Analytic continuation Definition of analytic continuation The Unique Continuation Theorem The uniqueness of analytic continuation Analytic continuation of the limit function of a power series Analytic continuation across a curve Global behaviour of holomorphic functions with nonisolated Wo-points Maximum Modulus Principle (Maximum Principle) 182 -

6 5.2.1 The basic Statement Holomorphic functions with constant modulus Mean Value Property of holomorphic functions Proof of the Maximum Modulus Principle A Maximum Modulus Principle for bounded domains The Minimum Modulus Principle (Minimum Principle) Entire functions Definition and basic properties Liouville's Theorem Functions of polynomial growth Fundamental Theorem of Algebra Statement of the problem Proofs of the Fundamental Theorem of Algebra using complex analysis Special case: Existence of the roots of complex numbers Argand's proof of the Fundamental Theorem of Algebra Additional proofs of the Fundamental Theorem of Algebra Factorization of polynomials Exercises to Chapter xi 6 Isolated singularities Classification Definition of isolated singularities Removable singularities Poles Essential singularities Laurent series Holomorphic functions in an annulus Holomorphic functions in a punctured disk Characterization by the principal part Meromorphic functions Behaviour at essential singularities Behaviour at infmity 211

7 6.7 Partial fractions of rational functions An application of the Division Algorithm Representation of rational functions by partial fractions Meromorphic functions on the Sphere Exercises to Chapter Homotopy Statement of the problem Homotopic curves Path independent line integrals Simply connected domains The concept of simple connectedness Cauchy's Integral Theorem in homotopy formulation Monodromy Theorem Solution of first order Systems A property of path independent line integrals Local Solution of first order Systems Global Solutions of first order Systems Conjugate Solutions Inversion of complex differentiation Morera's Theorem Potentials of vector fields The concept of a potential Some physical interpretations of vector fields Curl-free and source-free vector fields Exercises to Chapter Residue theory Statement of the problem Winding numbers The Integration of principal parts Termwise Integration A complex version of the Fundamental Theorem of Differential and Integral Calculus 247

8 8.3.3 Integration of meromorphic functions with first Order xiii poles Residue Theorem Calculation of residues The case of first order poles The case of poles of order fe > Determination of residues using Laurent series Exercises to Chapter Applications of residue calculus Total number of zeros and poles Representation by a boundary integral A proof of Fundamental Theorem of Algebra based on a boundary integral representation Rouche's Theorem Another proof of Fundamental Theorem of Algebra using Rouche's Theorem Evaluation of definite integrals Evaluation of integrals involving certain periodic functions between the limits 0 and 2-7T Evaluation of improper real integrals Integrals involving many-valued functions Sum of certain series Exercises to Chapter Mapping properties Continuously differentiable mappings Invertible linear mappings. The general case The exceptional case Calculation of the angle of rotation Orientation-preserving mappings Conformal mappings Conformal mappings by holomorphic functions Behaviour at zeros of the derivative Inversion of multivalent functions * 295

9 xiv Another proof of the local existence of the inverse function * Domain invariance * Behaviour of the Cauchy-Riemann System under conformal mappings * Conformal equivalence * Quasiconformal mappings * Examples of conformal mappings Some elementary conformal mappings The Möbius fractional linear transformations Mappings of the unit disk onto itself Complex plane onto itself Schwarz Reflection Principle Univalent functions * Definition and basic properties Bieberbach's conjecture Univalent functions outside the unit disk Proof of Bieberbach's conjecture for a Koebe's Covering Theorem Limits of Univalent functions Riemann's Mapping Theorem * Statement of the problem Outline of the proof; extremal problems in classes of holomorphic functions Proof of Riemann's Mapping Theorem Summary of the Solution of the main problems of Conformal Mappings Construction of flow lines * The level curves of real and imaginary parts of holomorphic functions Construction of curl-free and source-free vector fields in the plane Flow lines Examples of the construction of flow lines Exercises to Chapter

10 XV 11 Special functions Prescribed principal parts Prescribed zeros Infinite products * Statement of the the problem Infinite products of complex numbers Infinite products of complex- valued functions Derivatives of infinite products Weierstrass products * Statement of the problem Entire functions without zeros Weierstrass Primary Factors Preliminaries for the application of Weierstrass' Primary Factors The Weierstrass Factorization Theorem Some examples Gamma function * Definition of the gamma function Functional equation of the gamma function Some elementary properties of the gamma function An integral representation of the gamma function in the right half-plane A partial fraction representation of the gamma function The Riemann zeta function * Definition of Riemann's zeta function Connexion between zeta function and prime numbers Analytic continuation of the Riemann zeta function Relationship between gamma function and zeta function Elliptic functions * Weierstrass'zeta function Weierstrass' p-function Periods of meromorphic functions Properties of elliptic functions 400

11 XVI Construction of elliptic functions with prescribed principal parts Related topics Exercises to Chapter Boundary value problems Preliminaries Harmonie functions and the Dirichlet problem Maximum Principle The Poisson Integral Formula Derivation of Poisson's Integral Formula Construction of Solutions of the Laplace equation by Poisson's Integral Formula Preliminaries for the proof of Theorem Proof of Theorem Examples Poisson kernel and its conjugate Cauchy Type Integrals * Statement of the problem Cauchy Type Integrals with constant density Cauchy Type Integrals with Hölder-continuous density The Plemelj Formulae Desired holomorphic functions * The Schwarz problem Solution of the Schwarz problem in a disk The Riemann boundary value problem Exercises to Chapter References 455 Index 457

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