دانلود کتاب 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.ContentsPreface v1 Arithmetic and topology in the complex plane 11.1 Arithmetic of complex numbers .................. 11.2 Analytic geometry with complex terminology ........... 81.3 Topological notions. The compactified plane ........... 121.4 Curves, paths, length elements ................... 171.5 Branches of the argument. Index of a closed curve with respectto a point .............................. 221.6 Domains with regular boundary .................. 281.7 Exercises .............................. 352 Functions of a complex variable 392.1 Real variable polynomials, complex variable polynomials,rational functions .......................... 392.2 Complex exponential functions, logarithms and powers.Trigonometric functions ...................... 412.3 Power series ............................ 462.4 Differentiation of functions of a complex variable ......... 592.5 Analytic functions of a complex variable ............. 692.6 Real analytic functions and their complex extension ....... 762.7 Exercises .............................. 803 Holomorphic functions and differential forms 853.1 Complex line integrals ....................... 853.2 Line integrals, vector fields and differential 1-forms ....... 883.3 The fundamental theorem of complex calculus .......... 943.4 Green’s formula .......................... 993.5 Cauchy’s Theorem and applications ................ 1073.6 Classical theorems ......................... 1113.7 Holomorphic functions as vector fields and harmonic functions . 1243.8 Exercises .............................. 1314 Local properties of holomorphic functions 1364.1 Cauchy integral formula ...................... 1364.2 Analytic functions and holomorphic functions........... 1404.3 Analyticity of harmonic functions. Fourier series ......... 145x Contents4.4 Zeros of analytic functions. Principle of analytic continuation . . 1484.5 Local behavior of a holomorphic function. The open mappingtheorem ............................... 1534.6 Maximum principle. Cauchy’s inequalities. Liouville’s theorem . 1564.7 Exercises .............................. 1595 Isolated singularities of holomorphic functions 1645.1 Isolated singular points ....................... 1645.2 Laurent series expansion ...................... 1685.3 Residue of a function at an isolated singularity .......... 1745.4 Harmonic functions on an annulus ................. 1785.5 Holomorphic functions and singular functions at infinity ..... 1805.6 The argument principle ....................... 1835.7 Dependence of the set of solutions of an equation with respectto parameters ............................ 1885.8 Calculus of real integrals ...................... 1905.9 Exercises .............................. 2036 Homology and holomorphic functions 2076.1 Homology of chains and simply connected domains ....... 2076.2 Homological versions of Green’s formula and Cauchy’s theorem . 2116.3 The residue theorem and the argument principle in a homologicalversion ............................... 2196.4 Cauchy’s theorem for locally exact differential forms ....... 2206.5 Characterizations of simply connected domains .......... 2236.6 The first homology group of a domain and de Rham’s theorem.Homotopy ............................. 2256.7 Harmonic functions on n-connected domains ........... 2296.8 Exercises .............................. 2327 Harmonic functions 2367.1 Problems of classical physics and harmonic functions ...... 2367.2 Harmonic functions on domains of Rn ............... 2447.3 Newtonian and logarithmic potentials. Riesz’ decompositionformulae .............................. 2537.4 Maximum principle. Dirichlet and Neumann homogeneousproblems .............................. 2627.5 Green’s function. The Poisson kernel ............... 2647.6 Plane domains: specific methods of complex variables. Dirichletand Neumann problems in the unit disc .............. 2687.7 The Poisson equation in Rn .................... 279Contents xi7.8 The Poisson equation and the non-homogeneous Dirichletand Neumann problems in a domain of Rn ............ 2947.9 The solution of the Dirichlet and Neumann problems in the ball . 3027.10 Decomposition of vector fields ................... 3077.11 Dirichlet’s problem and conformal transformations ........ 3147.12 Dirichlet’s principle ........................ 3187.13 Exercises .............................. 3208 Conformal mapping 3268.1 Conformal transformations..................... 3268.2 Conformal mappings ........................ 3288.3 Homographic transformations ................... 3358.4 Automorphisms of simply connected domains........... 3438.5 Dirichlet’s problem and Neumann’s problem in the half plane . . 3478.6 Level curves ............................ 3498.7 Elementary conformal transformations .............. 3538.8 Conformal mappings of polygons ................. 3608.9 Conformal mapping of doubly connected domains ........ 3688.10 Applications of conformal mapping ................ 3718.11 Exercises .............................. 3799 The Riemann mapping theorem and Dirichlet’s problem 3839.1 Sequences of holomorphic or harmonic functions ......... 3839.2 Riemann’s theorem ......................... 3969.3 Green’s function and conformal mapping ............. 3989.4 Solution of Dirichlet’s problem in an arbitrary domain ...... 4039.5 Exercises .............................. 41110 Runge’s theorem and the Cauchy–Riemann equations 41510.1 Runge’s approximation theorems ................. 41510.2 Approximation of harmonic functions ............... 42310.3 Decomposition of meromorphic functions into simple elements . 42510.4 The non-homogeneous Cauchy–Riemann equations in the plane.The Cauchy integral ........................ 43610.5 The non-homogeneous Cauchy–Riemann equations in an open set.Weighted kernels .......................... 44110.6 The Dirichlet problem for the N@ operator .............. 45010.7 Exercises .............................. 455xii Contents11 Zeros of holomorphic functions 46011.1 Infinite products .......................... 46011.2 The Weierstrass factorization theorem ............... 46611.3 Interpolation by entire functions .................. 47311.4 Zeros of holomorphic functions and the Poisson equation .... 47711.5 Jensen’s formula .......................... 48111.6 Growth of a holomorphic function and distribution of the zeros . 48411.7 Entire functions of finite order ................... 48711.8 Ideals of the algebra of holomorphic functions .......... 49411.9 Exercises .............................. 50012 The complex Fourier transform 50412.1 The complex extension of the Fourier transform.First Paley–Wiener theorem .................... 50412.2 The Poisson formula ........................ 50912.3 Bandlimited functions. Second Paley–Wiener theorem ...... 51212.4 The Laplace transform ....................... 52112.5 Applications of the Laplace transform ............... 53312.6 Dirichlet series ........................... 54412.7 The Z-transform .......................... 54612.8 Exercises .............................. 550References 555Symbols 557Index 559