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

Contents xi

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