Overview

This book is written to be a convenient reference for the working scientist, student, or engineer who needs to know and use basic concepts in complex analysis. It is not a book of mathematical theory. It is instead a book of mathematical practice. All the basic ideas of complex analysis, as well as many typical applica tions, are treated. Since we are not developing theory and proofs, we have not been obliged to conform to a strict logical ordering of topics. Instead, topics have been organized for ease of reference, so that cognate topics appear in one place. Required background for reading the text is minimal: a good ground ing in (real variable) calculus will suffice. However, the reader who gets maximum utility from the book will be that reader who has had a course in complex analysis at some time in his life. This book is a handy com pendium of all basic facts about complex variable theory. But it is not a textbook, and a person would be hard put to endeavor to learn the subject by reading this book.

Full Product Details

Author:   Steven G. Krantz
Publisher:   Birkhauser Verlag AG
Imprint:   Birkhauser Verlag AG
ISBN:  

9783764340117

ISBN 10:   3764340118
Pages:   352
Publication Date:   August 1999
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Replaced By:   9780817640118
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

Part 1 The complex plane: complex arithmetic; the exponential and applications; holomorphic functions; the relationship of holomorphic and harmonic functions. Part 2 Complex line integrals: real and complex line integrals; complex differentiability and conformality; the Cauchy integral theorem and formula. Part 3 Applications of the Cauchy theory: the derivatives of a holomorphic function; the zeros of a holomorphic function. Part 4 Isolated singularities and Laurent series: the behaviour of a holomorphic function near an isolated singularity; expansion around singular points; examples of Laurent expansions; the calculus of residues; applications to the calculation of definite integrals and sums; mesomorphic functions and singularities at infinity. Part 5 The argument principle: counting zeros and poles; the local geometry of holomorphic functions; further results on the zeros of holomorphic functions; the maximum principle; the Schwarz lemma. Part 6 Holomorphic functions as geometric mappings: the idea of a conformal mapping; conformal mapping of the unit disc; linear fractional transformations; the Riemann mapping theorem; conformal mappings of annuli. Part 7 Harmonic functions: basic properties of harmonic functions; the maximum principle and the mean value property; the Poisson integral formula; regularity of harmonic functions; the Schwarz reflection principle; Harnack's principle; the Dirichlet problem and subharmonic functions; the general solution of the Dirichlet problem.Part 8 Infinite series and products: basic concepts concerning infinite sums and products; the Weierstrass factorization theorem; the theorems of Weierstrass and Mittag-Leffler; normal families. Part 9 Applications of infinite sums and products: Jensen's formula and an introduction to Blaschke products; the Hadamard Gap theorem; entire functions of finite order. Part 10 Analytic continuation: definition of an analytic function element; analytic continuation along a curve; the Monodromy theorem; the idea of a Riemann surface; Picard's theorem. Part 11 rational approximation theory: Runge's theorem; Mergelyan's theorem. Part 12 Special classes of holomorphic functions: Schlicht functions and the Bieberbach conjecture; extension to the boundary of conformal mappings; hardy spaces. Part 13 Special functions: the gamma and beta functions; Riemann's zeta functions; some counting functions an a few technical lemmas. Part 14 Applications that depend on conformal mappings: conformal mapping; application of conformal mapping to the Dirichlet problem; physical examples solved by means of conformal mapping; numerical techniques of conformal mapping. Part 15 Transform theory: introductory remarks; Fourier series; the Fourier transform; the Laplace transform; the z-transform. (Part contents).

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