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.
Contents 1 Preliminaries 1 1.1 The field of complex numbers 1 1.1.1 Introduction 1 1.1.2 Complex numbers as pairs of real numbers 1 1.1.3 Solvability of z 2 = -1 3 1.1.4 Definition of complex numbers using equivalence classes of polynomials 4 1.2 The complex plane 6 1.2.1 Geometrical Interpretation of complex numbers... 6 1.2.2 Absolute value. Conjugate complex numbers 6 1.2.3 Interpretation of complex numbers as vectors 7 1.2.4 Trigonometrie form of complex numbers 9 1.2.5 Geometrical Interpretation of the produet of complex numbers 9 1.2.6 Powers and roots of complex numbers 10 1.2.7 Some special sets in the complex plane 11 1.2.8 Examples 12 1.3 Metrie Spaces 14 1.3.1 The coneept of a metric space 14 1.3.2 Open sets 16 1.3.3 Convergence 18 1.3.4 Closed subsets 18 1.3.5 Definition of some more terms 20 1.3.6 Completeness 22 1.3.7 Compact sets 24 1.3.8 Coverings 25 1.3.9 Examples 27 1.4 Mappings and funetions. Continuity 28 vii
viii 1.4.1 Basic definitions 28 1.4.2 The extended complex plane. Spherical distance... 31 1.4.3 Limits of functions 34 1.4.4 Continuous functions 35 1.4.5 Curves 38 1.4.6 Connectivity 39 1.4.7 Sequences of functions 40 1.4.8 Infinite series 45 1.4.9 Power series 57 1.4.10 The complex exponential and trigonometric functions 67 1.4.11 Logarithm and its Riemann Surface 75 1.4.12 Remark on the concept of a topological space 80 1.4.13 Examples 81 1.5 Exercises to Chapter 1 82 2 The classical approach 87 2.1 Ordinary complex differentiation 87 2.1.1 Complex differentiability 87 2.1.2 Rules of differentiation 90 2.1.3 The Cauchy-Riemann System 90 2.1.4 Holomorphic functions 94 2.1.5 Differentiation of Power Series 96 2.1.6 Differentiation of inverse functions 100 2.2 Preliminaries of the Integral Calculus 102 2.2.1 Line integrals of real valued functions 102 2.2.2 The Green-Gauss Integral Theorem 104 2.2.3 Line integrals of complex valued functions 106 2.3 Complex Integral Theorems 112 2.3.1 Cauchy's Integral Theorem 113 2.3.2 Cauchy's Integral Formula 115 2.4 Exercises to Chapter 2 116
IX An alternative approach 121 3.1 Partial complex differentiations 121 3.1.1 Linearization of functions of one real variable 121 3.1.2 Linearization of functions depending on several real variables 121 3.1.3 Linearization of functions depending on one complex variable 122 3.1.4 Definition of partial complex derivatives 122 3.1.5 Differentiability rules for partial complex derivatives. 123 3.2 Complex Green-Gauss Integral Theorems * 123 3.3 Generalized Cauchy Integral Formula * 125 3.3.1 Application of complex version of the Green-Gauss Formula to functions having isolated singularities 125 3.3.2 The limit of the domain integral. Schmidt's Inequality 125 3.3.3 The limit of the line integral 128 3.3.4 An integral representation formula for continuously differentiable functions 128 3.4 The classical Cauchy Integral Formula * 129 3.4.1 Another approach to Cauchy's Integral Formula... 129 3.4.2 A second proof of Cauchy's Integral Theorem 129 3.5 Comparison * 130 3.5.1 Partial complex derivatives of functions having an ordinary complex derivative 130 3.5.2 Ordinary complex differentiability of Solutions of the Cauchy-Riemann System 130 3.6 Exercises to Chapter 3 131 Local properties 135 4.1 Existence of higher order derivatives 135 4.1.1 A method for proving local properties of holomorphic functions 135 4.1.2 The holomorphy of line integrals with respect to complex parameters 135 4.1.3 Cauchy's Integral Formula for the derivatives of a holomorphic function 137 4.2 Local power series representation 138 4.2.1 Power series representation of the Cauchy kernel... 138
X 4.2.2 Local power series for holomorphic functions and an integral representation of the coefficients 139 4.2.3 Cauchy's estimate of the coefficients 142 4.2.4 The power series of the product of two holomorphic functions 142 4.2.5 Division of power series 143 4.3 Distribution of zeros 144 4.4 The Weierstrass Convergence Theorem 145 4.4.1 Statement of the problem 145 4.4.2 Formulation and proof 145 4.4.3 Termwise differentiability 146 4.5 Connexion with plane Potential Theory 147 4.5.1 Holomorphic functions as Solutions of the Laplace equation 147 4.5.2 Representation of the Laplace Operator by partial complex differentiations 148 4.6 Complex Integral Theorems revisited * 148 4.6.1 Goursat's Theorem 149 4.6.2 G. Fichera's proof of the Goursat Theorem 152 4.6.3 A measure-theoretic approach to Cauchy's Integral Theorem 155 4.6.4 Consequences of Complex Integral Theorems under weaker assumptions 163 4.7 Exercises to Chapter 4 163 5 Global properties 169 5.1 Analytic continuation 169 5.1.1 Definition of analytic continuation 169 5.1.2 The Unique Continuation Theorem 170 5.1.3 The uniqueness of analytic continuation 171 5.1.4 Analytic continuation of the limit function of a power series 172 5.1.5 Analytic continuation across a curve 180 5.1.6 Global behaviour of holomorphic functions with nonisolated Wo-points 181 5.2 Maximum Modulus Principle (Maximum Principle) 182 -
5.2.1 The basic Statement 182 5.2.2 Holomorphic functions with constant modulus... 182 5.2.3 Mean Value Property of holomorphic functions... 183 5.2.4 Proof of the Maximum Modulus Principle 183 5.2.5 A Maximum Modulus Principle for bounded domains 184 5.2.6 The Minimum Modulus Principle (Minimum Principle) 184 5.3 Entire functions 186 5.3.1 Definition and basic properties 186 5.3.2 Liouville's Theorem 187 5.3.3 Functions of polynomial growth 187 5.4 Fundamental Theorem of Algebra 188 5.4.1 Statement of the problem 188 5.4.2 Proofs of the Fundamental Theorem of Algebra using complex analysis 189 5.4.3 Special case: Existence of the roots of complex numbers 192 5.4.4 Argand's proof of the Fundamental Theorem of Algebra 193 5.4.5 Additional proofs of the Fundamental Theorem of Algebra 195 5.4.6 Factorization of polynomials 195 5.5 Exercises to Chapter 5 195 xi 6 Isolated singularities 199 6.1 Classification 199 6.1.1 Definition of isolated singularities 199 6.1.2 Removable singularities 199 6.1.3 Poles 201 6.1.4 Essential singularities 202 6.2 Laurent series 202 6.2.1 Holomorphic functions in an annulus 202 6.2.2 Holomorphic functions in a punctured disk 205 6.3 Characterization by the principal part 206 6.4 Meromorphic functions 208 6.5 Behaviour at essential singularities 210 6.6 Behaviour at infmity 211
6.7 Partial fractions of rational functions 213 6.7.1 An application of the Division Algorithm 213 6.7.2 Representation of rational functions by partial fractions 214 6.8 Meromorphic functions on the Sphere 216 6.9 Exercises to Chapter 6 217 Homotopy 223 7.1 Statement of the problem 223 7.2 Homotopic curves 225 7.3 Path independent line integrals 227 7.4 Simply connected domains 229 7.4.1 The concept of simple connectedness 229 7.4.2 Cauchy's Integral Theorem in homotopy formulation. 229 7.4.3 Monodromy Theorem 230 7.5 Solution of first order Systems 232 7.5.1 A property of path independent line integrals 232 7.5.2 Local Solution of first order Systems 233 7.5.3 Global Solutions of first order Systems 234 7.6 Conjugate Solutions 234 7.7 Inversion of complex differentiation 235 7.8 Morera's Theorem 237 7.9 Potentials of vector fields 238 7.9.1 The concept of a potential 238 7.9.2 Some physical interpretations of vector fields 239 7.9.3 Curl-free and source-free vector fields 240 7.10 Exercises to Chapter 7 241 Residue theory 245 8.1 Statement of the problem 245 8.2 Winding numbers 245 8.3 The Integration of principal parts 246 8.3.1 Termwise Integration 246 8.3.2 A complex version of the Fundamental Theorem of Differential and Integral Calculus 247
8.3.3 Integration of meromorphic functions with first Order xiii poles 249 8.4 Residue Theorem 249 8.5 Calculation of residues 251 8.5.1 The case of first order poles 252 8.5.2 The case of poles of order fe > 2 253 8.5.3 Determination of residues using Laurent series... 253 8.6 Exercises to Chapter 8 254 9 Applications of residue calculus 257 9.1 Total number of zeros and poles 257 9.1.1 Representation by a boundary integral 257 9.1.2 A proof of Fundamental Theorem of Algebra based on a boundary integral representation 258 9.1.3 Rouche's Theorem 259 9.1.4 Another proof of Fundamental Theorem of Algebra using Rouche's Theorem 259 9.2 Evaluation of definite integrals 260 9.2.1 Evaluation of integrals involving certain periodic functions between the limits 0 and 2-7T 260 9.2.2 Evaluation of improper real integrals 263 9.2.3 Integrals involving many-valued functions 273 9.3 Sum of certain series 277 9.4 Exercises to Chapter 9 280 10 Mapping properties 287 10.1 Continuously differentiable mappings 287 10.1.1 Invertible linear mappings. The general case 287 10.1.2 The exceptional case 290 10.1.3 Calculation of the angle of rotation 291 10.1.4 Orientation-preserving mappings 292 10.2 Conformal mappings 293 10.2.1 Conformal mappings by holomorphic functions... 293 10.2.2 Behaviour at zeros of the derivative 294 10.2.3 Inversion of multivalent functions * 295
xiv 10.2.4 Another proof of the local existence of the inverse function * 296 10.2.5 Domain invariance * 296 10.2.6 Behaviour of the Cauchy-Riemann System under conformal mappings * 297 10.2.7 Conformal equivalence * 297 10.2.8 Quasiconformal mappings * 297 10.3 Examples of conformal mappings 299 10.3.1 Some elementary conformal mappings 299 10.3.2 The Möbius fractional linear transformations 302 10.3.3 Mappings of the unit disk onto itself 307 10.3.4 Complex plane onto itself 313 10.3.5 Schwarz Reflection Principle 314 10.4 Univalent functions * 316 10.4.1 Definition and basic properties 316 10.4.2 Bieberbach's conjecture 316 10.4.3 Univalent functions outside the unit disk 317 10.4.4 Proof of Bieberbach's conjecture for a 2 318 10.4.5 Koebe's Covering Theorem 320 10.4.6 Limits of Univalent functions 320 10.5 Riemann's Mapping Theorem * 321 10.5.1 Statement of the problem 321 10.5.2 Outline of the proof; extremal problems in classes of holomorphic functions 322 10.5.3 Proof of Riemann's Mapping Theorem 322 10.5.4 Summary of the Solution of the main problems of Conformal Mappings 326 10.6 Construction of flow lines * 328 10.6.1 The level curves of real and imaginary parts of holomorphic functions 328 10.6.2 Construction of curl-free and source-free vector fields in the plane 328 10.6.3 Flow lines 329 10.6.4 Examples of the construction of flow lines 330 10.7 Exercises to Chapter 10 332
XV 11 Special functions 339 11.1 Prescribed principal parts 339 11.2 Prescribed zeros 346 11.3 Infinite products * 348 11.3.1 Statement of the the problem 348 11.3.2 Infinite products of complex numbers 349 11.3.3 Infinite products of complex- valued functions 355 11.3.4 Derivatives of infinite products 359 11.4 Weierstrass products * 362 11.4.1 Statement of the problem 362 11.4.2 Entire functions without zeros 363 11.4.3 Weierstrass Primary Factors 364 11.4.4 Preliminaries for the application of Weierstrass' Primary Factors 366 11.4.5 The Weierstrass Factorization Theorem 367 11.4.6 Some examples 368 11.5 Gamma function * 369 11.5.1 Definition of the gamma function 369 11.5.2 Functional equation of the gamma function 372 11.5.3 Some elementary properties of the gamma function.. 372 11.5.4 An integral representation of the gamma function in the right half-plane 375 11.5.5 A partial fraction representation of the gamma function 379 11.6 The Riemann zeta function * 381 11.6.1 Definition of Riemann's zeta function 381 11.6.2 Connexion between zeta function and prime numbers 382 11.6.3 Analytic continuation of the Riemann zeta function. 383 11.6.4 Relationship between gamma function and zeta function 386 11.7 Elliptic functions * 388 11.7.1 Weierstrass'zeta function 388 11.7.2 Weierstrass' p-function 390 11.7.3 Periods of meromorphic functions 393 11.7.4 Properties of elliptic functions 400
XVI 11.7.5 Construction of elliptic functions with prescribed principal parts 406 11.7.6 Related topics 408 11.8 Exercises to Chapter 11 416 12 Boundary value problems 425 12.1 Preliminaries 425 12.1.1 Harmonie functions and the Dirichlet problem... 425 12.1.2 Maximum Principle 426 12.2 The Poisson Integral Formula 428 12.2.1 Derivation of Poisson's Integral Formula 428 12.2.2 Construction of Solutions of the Laplace equation by Poisson's Integral Formula 430 12.2.3 Preliminaries for the proof of Theorem 101 431 12.2.4 Proof of Theorem 101 432 12.2.5 Examples 433 12.2.6 Poisson kernel and its conjugate 435 12.3 Cauchy Type Integrals * 436 12.3.1 Statement of the problem 436 12.3.2 Cauchy Type Integrals with constant density 437 12.3.3 Cauchy Type Integrals with Hölder-continuous density 440 12.3.4 The Plemelj Formulae 444 12.4 Desired holomorphic functions * 447 12.4.1 The Schwarz problem 447 12.4.2 Solution of the Schwarz problem in a disk 448 12.4.3 The Riemann boundary value problem 449 12.5 Exercises to Chapter 12 450 References 455 Index 457