Complex Analysis

The theory of functions of a complex variable is a central theme in mathematical analysis that has links to several branches of mathematics. Understanding the basics of the theory is necessary for anyone who wants to have a general mathematical training or for anyone who wants to use mathematics in applied sciences or technology. The book presents the basic theory of analytic functions of a complex variable and their points of contact with other parts of mathematical analysis. This results in some new approaches to a number of topics when compared to the current literature on the subject. Some issues covered are: a real version of the Cauchy–Goursat theorem, theorems of vector analysis with weak regularity assumptions, an approach to the concept of holomorphic functions of real variables, Green’s formula with multiplicities, Cauchy’s theorem for locally exact forms, a study in parallel of Poisson’s equation and the inhomogeneous Cauchy–Riemann equations, the relationship between Green’s function and conformal mapping, the connection between the solution of Poisson’s equation and zeros of holomorphic functions, and the Whittaker–Shannon theorem of information theory. The text can be used as a manual for complex variable courses of various levels and as a reference book. The only prerequisites for reading it is a working knowledge of the topology of the plane and the differential calculus for functions of several real variables. A detailed treatment of harmonic functions also makes the book useful as an introduction to potential theory. Contents Preface v 1 Arithmetic and topology in the complex plane 1 1.1 Arithmetic of complex numbers .................. 1 1.2 Analytic geometry with complex terminology ........... 8 1.3 Topological notions. The compactified plane ........... 12 1.4 Curves, paths, length elements ................... 17 1.5 Branches of the argument. Index of a closed curve with respect to a point .............................. 22 1.6 Domains with regular boundary .................. 28 1.7 Exercises .............................. 35 2 Functions of a complex variable 39 2.1 Real variable polynomials, complex variable polynomials, rational functions .......................... 39 2.2 Complex exponential functions, logarithms and powers. Trigonometric functions ...................... 41 2.3 Power series ............................ 46 2.4 Differentiation of functions of a complex variable ......... 59 2.5 Analytic functions of a complex variable ............. 69 2.6 Real analytic functions and their complex extension ....... 76 2.7 Exercises .............................. 80 3 Holomorphic functions and differential forms 85 3.1 Complex line integrals ....................... 85 3.2 Line integrals, vector fields and differential 1-forms ....... 88 3.3 The fundamental theorem of complex calculus .......... 94 3.4 Green’s formula .......................... 99 3.5 Cauchy’s Theorem and applications ................ 107 3.6 Classical theorems ......................... 111 3.7 Holomorphic functions as vector fields and harmonic functions . 124 3.8 Exercises .............................. 131 4 Local properties of holomorphic functions 136 4.1 Cauchy integral formula ...................... 136 4.2 Analytic functions and holomorphic functions........... 140 4.3 Analyticity of harmonic functions. Fourier series ......... 145x Contents 4.4 Zeros of analytic functions. Principle of analytic continuation . . 148 4.5 Local behavior of a holomorphic function. The open mapping theorem ............................... 153 4.6 Maximum principle. Cauchy’s inequalities. Liouville’s theorem . 156 4.7 Exercises .............................. 159 5 Isolated singularities of holomorphic functions 164 5.1 Isolated singular points ....................... 164 5.2 Laurent series expansion ...................... 168 5.3 Residue of a function at an isolated singularity .......... 174 5.4 Harmonic functions on an annulus ................. 178 5.5 Holomorphic functions and singular functions at infinity ..... 180 5.6 The argument principle ....................... 183 5.7 Dependence of the set of solutions of an equation with respect to parameters ............................ 188 5.8 Calculus of real integrals ...................... 190 5.9 Exercises .............................. 203 6 Homology and holomorphic functions 207 6.1 Homology of chains and simply connected domains ....... 207 6.2 Homological versions of Green’s formula and Cauchy’s theorem . 211 6.3 The residue theorem and the argument principle in a homological version ............................... 219 6.4 Cauchy’s theorem for locally exact differential forms ....... 220 6.5 Characterizations of simply connected domains .......... 223 6.6 The first homology group of a domain and de Rham’s theorem. Homotopy ............................. 225 6.7 Harmonic functions on n-connected domains ........... 229 6.8 Exercises .............................. 232 7 Harmonic functions 236 7.1 Problems of classical physics and harmonic functions ...... 236 7.2 Harmonic functions on domains of Rn ............... 244 7.3 Newtonian and logarithmic potentials. Riesz’ decomposition formulae .............................. 253 7.4 Maximum principle. Dirichlet and Neumann homogeneous problems .............................. 262 7.5 Green’s function. The Poisson kernel ............... 264 7.6 Plane domains: specific methods of complex variables. Dirichlet and Neumann problems in the unit disc .............. 268 7.7 The Poisson equation in Rn .................... 279Contents xi 7.8 The Poisson equation and the non-homogeneous Dirichlet and Neumann problems in a domain of Rn ............ 294 7.9 The solution of the Dirichlet and Neumann problems in the ball . 302 7.10 Decomposition of vector fields ................... 307 7.11 Dirichlet’s problem and conformal transformations ........ 314 7.12 Dirichlet’s principle ........................ 318 7.13 Exercises .............................. 320 8 Conformal mapping 326 8.1 Conformal transformations..................... 326 8.2 Conformal mappings ........................ 328 8.3 Homographic transformations ................... 335 8.4 Automorphisms of simply connected domains........... 343 8.5 Dirichlet’s problem and Neumann’s problem in the half plane . . 347 8.6 Level curves ............................ 349 8.7 Elementary conformal transformations .............. 353 8.8 Conformal mappings of polygons ................. 360 8.9 Conformal mapping of doubly connected domains ........ 368 8.10 Applications of conformal mapping ................ 371 8.11 Exercises .............................. 379 9 The Riemann mapping theorem and Dirichlet’s problem 383 9.1 Sequences of holomorphic or harmonic functions ......... 383 9.2 Riemann’s theorem ......................... 396 9.3 Green’s function and conformal mapping ............. 398 9.4 Solution of Dirichlet’s problem in an arbitrary domain ...... 403 9.5 Exercises .............................. 411 10 Runge’s theorem and the Cauchy–Riemann equations 415 10.1 Runge’s approximation theorems ................. 415 10.2 Approximation of harmonic functions ............... 423 10.3 Decomposition of meromorphic functions into simple elements . 425 10.4 The non-homogeneous Cauchy–Riemann equations in the plane. The Cauchy integral ........................ 436 10.5 The non-homogeneous Cauchy–Riemann equations in an open set. Weighted kernels .......................... 441 10.6 The Dirichlet problem for the N @ operator .............. 450 10.7 Exercises .............................. 455xii Contents 11 Zeros of holomorphic functions 460 11.1 Infinite products .......................... 460 11.2 The Weierstrass factorization theorem ............... 466 11.3 Interpolation by entire functions .................. 473 11.4 Zeros of holomorphic functions and the Poisson equation .... 477 11.5 Jensen’s formula .......................... 481 11.6 Growth of a holomorphic function and distribution of the zeros . 484 11.7 Entire functions of finite order ................... 487 11.8 Ideals of the algebra of holomorphic functions .......... 494 11.9 Exercises .............................. 500 12 The complex Fourier transform 504 12.1 The complex extension of the Fourier transform. First Paley–Wiener theorem .................... 504 12.2 The Poisson formula ........................ 509 12.3 Bandlimited functions. Second Paley–Wiener theorem ...... 512 12.4 The Laplace transform ....................... 521 12.5 Applications of the Laplace transform ............... 533 12.6 Dirichlet series ........................... 544 12.7 The Z-transform .......................... 546 12.8 Exercises .............................. 550 References 555 Symbols 557 Index 559