| 复分析(也称为复变函数论)是数学的重要分支，产生于18世纪，在19世纪获得了全面的发展，其研究领域在20世纪初得到了更进一步的拓广。它不仅在空气动力学、流体力学、电学、热学、理论物理等自然科学领域有着广泛的应用，而且已经深入到微分方程、积分方程、概率论和数论等其他数学分支中，对它们的发展影响极大。它的基础内容已成为理工科很多专业的必修课程。 本书出自菲尔兹奖和沃尔夫奖双奖得主，日本最伟大的数学家之一小平邦彦之手，图文并茂，强调理论的几何直觉。例题和习题(附有解答)都非常丰富，是一本经典的复分析著作，既可以作为课堂教材，也可以供研究参考。 |
| 小平邦彦，20世纪日本最伟大的数学家之一，他是迄今为止为数不多的既获得菲尔兹奖(1954年)、又获得沃尔夫奖(1985年)的数学家。1957年被日本政府授予文化勋章。他是日本学士院院士、美国科学院和德国哥廷根科学院外籍院士。先后在美国普林斯顿高等研究中心、哈佛大学、约翰·霍普金斯大学、斯坦福大学、日本东京大学等任教授。他在调和积分理论、代数几何学和复解析几何学等诸多领域做出了卓越的贡献，著作有《微积分入门》(卷Ⅰ和卷Ⅱ)、《复分析》、《复流形理论》等。 |
| 1 Holomorphic functions 1 1.1 Holomorphic functions 1 a. The complex plane 1 b. Functions of a complex variable 5 c. Holomorphic functions 9 1.2 Power series 15 a. Series whose terms are functions 15 b. Power series 16 1.3 Integrals 24 a. Curves 25 b. Integrals 30 c. Cauchy's integral formula for circles 35 d. Power series expansions 43 1.4 Properties ofholomorphic functions 47 a. mth-order derivatives 47 b. Limits of sequences of holomorphic fimctions 49 c. The Mean Value Theorem and the maximum principle 51 d. Isolated singularities 52 e. Entire functions 58 2 Cauchy's Theorem 60 2.1 Piecewise smooth curves 60 a. Smooth Jordan curves 60 b. Boundaries of bounded closed regions 65 2.2 Cellular decomposition 74 a. Calls 74 b. Cellular decomposition 80 2.3 Cauchy's Theorem 101 a. Cauchy's Theorem 101 b. Cauchy's integral formula 104 c. Residues 106 d. Evaluation of definite integrals 109 2.4 Differentiability and homology 113 3 Conformal mappings 117 3.1 Conformal mappings 117 3.2 The Riemann sphere 132 a. The Riemann sphere 132 b. Holomorphic functions with an isolated singularity at ∞ 135 c. Local coordinates 137 d. Homogeneous coordinates 138 3.3 Linear fractional transformations 139 a. Linear fractional transformations 139 b. Cross ratio 142 c. Elementary conformal mappings 148 4 Analytic continuation 153 4.1 Analytic continuation 153 a. Analytic continuation 153 b. Analytic continuation by expansion in power series 157 4.2 Analytic continuation along curves 160 4.3 Analytic continuation by integrals 180 4.4 Cauchy's Theorem (continued) 190 5 Riemann's Mapping Theorem 200 5.1 Riemann's Mapping Theorem 200 5.2 Correspondence of boundaries 214 5.3 The principle of reflection 214 a. The principle of reflection 224 b. Modular functions 233 c. Picard's Theorem 241 d. The Schwarz-Christoffel formula 241 6 Riemann surfaces 247 6.1 Differential forms 247 a. Differential forms 247 b. Line integrals 249 c. Harmonic forms 257 d. Harmonic functions 258 6.2 Riemann surfaces 260 a. Hausdorff spaces 260 b. Definition of Riemann surfaces 263 6.3 Differential forms on a Riemann surface 268 a. Differential forms 268 b. Line integrals 272 c. Locally finite open coverings 275 d. Partition of unity 278 e. Green's Theorem 284 6.4 Dirichlet's Principle 294 a. Inner product and norm 294 b. Dirichlet's Principle 299 c. Analytic fimctions 314 7 The structure of Riemann surfaces 319 7.1 Planar Riemann surfaces 319 a. Planar Riemann surfaces 319 b. Simply connected Riemann surfaces 329 c. Multiply connected regions 333 7.2 Compact Riemann surfaces 340 a. Cohomology groups 340 b. Structure of compact Riemann surfaces 344 c. Homology groups 360 8 Analytic functions on a closed Riemann surface 376 8.1 Abelian differentials of the first kind 376 a. Harmonic 1-forms of the first kind 376 b. Abelian differential of the first kind 379 8.2 Abelian differentials of the second and third kind 379 a. Meromorphic functions 379 b. Abelian differentials of the second and third kind 380 8.3 The Riemann-Roch Theorem 381 a. Existence Theorem 381 b. The Riemann-Roch Theorem 382 8.4 Abel's Theorem 389 a. Existence Theorem 389 b. Abel's Theorem 391 Problems 394 Index 401 1 Holomorphic functions 1 1.1 Holomorphic functions 1 a. The complex plane 1 b. Functions of a complex variable 5 c. Holomorphic functions 9 1.2 Power series 15 a. Series whose terms are functions 15 b. Power series 16 1.3 Integrals 24 a. Curves 25 b. Integrals 30 c. Cauchy's integral formula for circles 35 d. Power series expansions 43 1.4 Properties ofholomorphic functions 47 a. mth-or |
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