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TRANSCRIPT
Joaquim Bruna Julia Cufi
· Complex Analysis
Translated from the Catalan by lgnacio Monreal
6 uropean ~thematical Society
Contents
Preface v
1 Arithmetic and topology in the complex plane 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.
Trigonometrie 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 145
x 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 Harmonie 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 . . . . . . . 5.8 Calculus of real integrals . 5. 9 Exercises . . . . . . . . .
188 190 203
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
6.4 6.5 6.6
6.7 6.8
version . . .... .. .. . .... . ........ . Cauchy's theorem for locally exact differential forms ..... . Characterizations of simply connected domains . . . . . . . . . The first homology group of a domain and de Rham's theorem. Homotopy ................. . Harmonie functions on n-connected domains Exercises ...
219 220 223
225 229 232
7 Harmonie functions 236 7 .1 Problems of classical physics and harmonic functions 236 7 .2 Harmonie 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 . . . • . . . . • . . . . • . . • . . • 279
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7 .8 Tue Poisson equation and the non-homogeneous Dirichlet and Neumann problems in a domain of IRn • • • • • • • • 294
7.9 Tue solution of the Dirichlet and Neumann problems in the ball 302
7 .10 Decomposition of vector fields . . . . . . . . . . . 7.11 Dirichlet's problem and confonnal transfonnations
7.12 Dirichlet's principle 7 .13 Exercises . . .
8 Confonnal mapping 8.1 Confonnal transfonnations .
8.2 Confonnal mappings . . . .
307 314
318
320
326 326
328 8.3 Homographic transfonnations 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 confonnal transfonnations . . . . . . 353 8.8 Confonnal mappings of polygons . . . . . . . . . 360
8.9 Confonnal mapping of doubly connected domains
8.10 Applications of confonnal mapping 8.11 Exercises . . . . . . . . . . . . . . . . . . . . . .
9 The Riemann mapping theorem and Dirichlet's problem 9.1 Sequences ofholomorphic or hannonic functions .
9.2 Riemann's theorem ........ .. ........ . 9.3 Green's function and confonnal mapping ...... .
9.4 Solution of Dirichlet's problem in an arbitrary domain 9.5 Exercises ..................... .
10 Runge's theorem and the Cauchy-Riemann equations 10.1 Runge's approximation theorems ........ .
10.2 Approximation of hannonic functions . . . . . . . . . . . . . .
10.3 Decomposition of meromorphic functions into simple elements 10.4 Tue non-homogeneous Cauchy-Riemann equations in the plane.
Tue Cauchy integral . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Tue non-homogeneous Cauchy-Riemann equations in an open set.
Weighted kemels . . . . . . . . . . . . .
10.6 Tue Dirichlet problem for the d Operator . 10.7 Exercises ... ............. .
368 371 379
383
383 396
398
403 411
415 415
423
425
436
441
450 455
xii Contents
11 Zeros of bolomorphic functions 460 11.1 Infinite products . . . . . . . . . . . . 460 11.2 Tue Weierstrass factorization theorem . 466 11.3 Interpolation by entire functions . . . . 473 11.4 Zeros of holomorphic functions and the Poisson equation 4 77 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 Tue complex extension of the Fourier transform.
First Paley-Wiener theorem . . . . . . . . . . . 504 12.2 Tue Poisson formula . . . . . . . . . . . . . . . 509 12.3 Bandlimited functions. Second Paley-Wiener theorem 512 12.4 Tue Laplace transform . . . . . . . . . 521 12.5 Applications of the Laplace transform . 533 12.6 Dirichlet series . 544 12.7 Tue Z-transform 546 12.8 Exercises 550
References 555
Symbols 557
Index 559