Second Edition. This famous work is a textbook that emphasizes the conceptual and historical continuity of analytic function theory. The second volume broadens from a textbook to a textbook-treatise, covering the ``canonical'' topics (including elliptic functions, entire and meromorphic functions, as well as conformal mapping, etc.) and other topics nearer the expanding frontier of analytic function theory. In the latter category are the chapters on majorization and on functions holomorphic in a half-plane.
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outstanding, rigorous classical treatment of complex variables,
This review is from: Analytic Function Theory, Volume I (AMS Chelsea Publishing) (Hardcover)
This book is by an outstanding master of the subject and a fine writer in the classical style. Hille treats the subject in detail, with many examples and historical references, but does not dumb the topic down, or render it soft. His treatment is more complete than many others for the serious beginner, since he proves such theorems as the one that having continuous partials satisfying the Cauchy Riemann equations implies holomorphicity, instead of just referring for this to the real variable case, which the student probably has never seen proved. He even proves a polygonal version of the jordan curve theorem in an appendix. His discussion gives the student a concrete feel for the subject by starting with serious examples such as fractional linear transformations, before going into theoretical development of integrals and power series. all that in volume I. Volume II is really thorough with a proof of the big Picard theorem, one of few places it is found in textbook form, and even some discussion of riemann surfaces of algebraic curves. this book is a great bargain and of high quality for one who wants a thorough serious treatment of analytic functions, from scratch.
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91
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- Theory of Functions of a Complex Variable (3 Vols. in 1) by A. I. Markushevich on page 379, Back Matter (1), and Back Matter (2)
- Chaos and Integrability in Nonlinear Dynamics: An Introduction by Michael Tabor on page 336, and page 357
- Mathematical and Statistical Methods for Genetic Analysis by Kenneth Lange on page 57, and page 316
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