Complex Analysis (Universitext)

The guiding principle of this presentation of ``Classical Complex Analysis'' is to proceed as quickly as possible to the central results while using a small number of notions and concepts from other fields. Thus the prerequisites for understanding this book are minimal; only elementary facts of calculus and algebra are required. The first four chapters cover the essential core of complex analysis: - differentiation in C (including elementary facts about conformal mappings) - integration in C (including complex line integrals, Cauchy's Integral Theorem, and the Integral Formulas) - sequences and series of analytic functions, (isolated) singularities, Laurent series, calculus of residues - construction of analytic functions: the gamma function, Weierstrass' Factorization Theorem, Mittag-Leffler Partial Fraction Decomposition, and -as a particular highlight- the Riemann Mapping Theorem, which characterizes the simply connected domains in C. Further topics included are: - the theory of elliptic functions based on the model of K. Weierstrass (with an excursions to older approaches due to N.H. Abel and C.G.J. Jacobi using theta series) - an introduction to the theory of elliptic modular functions and elliptic modular forms - the use of complex analysis to obtain number theoretical results - a proof of the Prime Number Theorem with a weak form of the error term. The book is especially suited for graduated students in mathematics and advanced undergraduated students in mathematics and other sciences. Motivating introductions, more than four hundred exercises of all levels of difficulty with hints or solutions, historical annotations, and over 120 figures make the overall presentation very attractive. The structure of the text, including abstracts beginning each chapter and highlighting of the main results, makes this book very appropriate for self-guided study and an indispensable aid in preparing for tests. This English edition is based on the fourth forthcoming German edition. This book is a translation of the forthcoming fourth edition of our German A book “Funktionentheorie I” (Springer 2005). The translation and the LT X E?les have been produced by Dan Fulea. He also made a lot of suggestions for improvement which in?uenced the English version of the book. It is a pleasure for us to express to him our thanks. We also want to thank our colleagues Diarmuid Crowley, Winfried Kohnen and J ̈ org Sixt for useful suggestions c- cerning the translation. Over the years, a great number of students, friends, and colleagues have c- tributed many suggestions and have helped to detect errors and to clear the text. The many new applications and exercises were completed in the last decade to also allow a partial parallel approach using computer algebra systems and graphictools,whichmayhaveafruitful,powerfulimpactespeciallyincomplex analysis. Last but not least, we are indebted to Clemens Heine (Springer, Heidelberg), who revived our translation project initially started by Springer, New York, and brought it to its?nal stage. The idea of this book is to give an extensive description of the classical complex analysis, here "classical" means roughly that sheaf theoretical and cohomological methods are omitted.

The first four chapters cover the essential core of complex analysis presenting their fundamental results. After this standard material, the authors step forward to elliptic functions and to elliptic modular functions including a taste of all most beautiful results of this field. The book is rounded by applications to analytic number theory including distinguished pearls of this fascinating subject as for instance the Prime Number Theorem. Great importance is attached to completeness, all needed notions are developed, only minimal prerequisites (elementary facts of calculus and algebra) are required.

More than 400 exercises including hints for solutions and many figures make this an attractive, indispensable book for students who would like to have a sound introduction to classical complex analysis.