Complex Analysis, Second Edition (Universitext)

• Publisher:   Springer
• Number Of Pages:   532
• Publication Date:   2009-05-01
• ISBN-10 / ASIN:   3540939822
• ISBN-13 / EAN:   9783540939825

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.

For the second edition the authors have revised the text carefully.

This book was originally written in German, and the German version is just incredible: a real gem. Good reason to translate it! Unfortunately, this is one of the worst translations from German I have seen. Some of it is just awkward grammar, which the reader may be able to ignore. But, there are also some words and phrases which are translated incorrectly.
For example "Paragraphen" in German does not mean paragraph in English, it means section. But in this book it is translated as paragraph. Try looking for something at the end of a paragraph or in the previous paragraph, when you should actually be looking at the end of the section or in the previous section. An example of this can be found in the explanation of the addition theorem for complex exponents (p. 27). The English text claims there is a remark concerning this at the end of the "paragraph." The paragraph ends and there is no remark. Turn to the end of the section (p. 31) and you will find the remark just above the exercises.
My advice is, if you can read German, get the German version! If you can't read German, you can still get the English version, but you will have to be very patient with the mistakes incurred in the translation (not to be found in the German original). If you own this book, you should systematically go through it and replace "paragraph" everywhere with "section." Most of the other translation mistakes can be figured out by context.

I was truly delighted to find this text. It starts off with ordinary complex analysis at the level of sophomore undergraduate students and proceeds well into graduate-level complex analysis (analytic number theory, elliptic functions, abels theorem, etc). The 'advanced' results are shown using standard methods, so it was a great way for someone who learned the nuts and bolts of contour integration to move into theta functions, the prime number theorem, etc etc...fun stuff.

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