Complex Variables: Introduction and Applications

by Mark J. Ablowitz , Athanassios S. Fokas

• ISBN13: 9780521534291
• ISBN10: 0521534291
• eBook ISBN(s): 9780511074288
• Edition: 2nd
• Format: Paperback
• Copyright: 2003-04-28
• Publisher: Cambridge University Press
• More Book Details
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Summary

Complex variables provide powerful methods for attacking problems that can be very difficult to solve in any other way, and it is the aim of this book to provide a thorough grounding in these methods and their application. Part I of this text provides an introduction to the subject, including analytic functions, integration, series, and residue calculus and also includes transform methods, ODEs in the complex plane, and numerical methods. Part II contains conformal mappings, asymptotic expansions, and the study of Riemann-Hilbert problems. The authors provide an extensive array of applications, illustrative examples and homework exercises. This new edition has been improved throughout and is ideal for use in undergraduate and introductory graduate level courses in complex variables.

Table of Contents

Preface	xi
Part I Fundamentals and Techniques of Complex Function Theory	1	(308)
Complex Numbers and Elementary Functions	3	(29)
Complex Numbers and Their Properties	3	(5)
Elementary Functions and Stereographic Projections	8	(12)
Elementary Functions	8	(7)
Stereographic Projections	15	(5)
Limits, Continuity, and Complex Differentiation	20	(6)
Elementary Applications to Ordinary Differential Equations	26	(6)
Analytic Functions and Integration	32	(77)
Analytic Functions	32	(14)
The Cauchy--Riemann Equations	32	(8)
Ideal Fluid Flow	40	(6)
Multivalued Functions	46	(15)
More Complicated Multivalued Functions and Riemann Surfaces	61	(9)
Complex Integration	70	(11)
Cauchy's Theorem	81	(10)
Cauchy's Integral Formula, Its ∂ Generalization and Consequences	91	(14)
Cauchy's Integral Formula and Its Derivatives	91	(4)
Liouville, Morera, and Maximum-Modulus Theorems	95	(3)
Generalized Cauchy Formula and ∂ Derivatives	98	(7)
Theoretical Developments	105	(4)
Sequences, Series, and Singularities of Complex Functions	109	(97)
Definitions and Basic Properties of Complex Sequences, Series	109	(5)
Taylor Series	114	(13)
Laurent Series	127	(10)
Theoretical Results for Sequences and Series	137	(7)
Singularities of Complex Functions	144	(14)
Analytic Continuation and Natural Barriers	152	(6)
Infinite Products and Mittag--Leffler Expansions	158	(16)
Differential Equations in the Complex Plane: Painleve Equations	174	(22)
Computational Methods	196	(10)
Laurent Series	196	(2)
Differential Equations	198	(8)
Residue Calculus and Applications of Contour Integration	206	(103)
Cauchy Residue Theorem	206	(11)
Evaluation of Certain Definite Integrals	217	(20)
Principal Value Integrals and Integrals with Branch Points	237	(22)
Principal Value Integrals	237	(8)
Integrals with Branch Points	245	(14)
The Argument Principle, Rouche's Theorem	259	(8)
Fourier and Laplace Transforms	267	(18)
Applications of Transforms to Differential Equations	285	(24)
Part II Applications of Complex Function Theory	309	(318)
Conformal Mappings and Applications	311	(100)
Introduction	311	(1)
Conformal Transformations	312	(5)
Critical Points and Inverse Mappings	317	(5)
Physical Applications	322	(19)
Theoretical Considerations -- Mapping Theorems	341	(4)
The Schwarz--Christoffel Transformation	345	(21)
Bilinear Transformations	366	(16)
Mappings Involving Circular Arcs	382	(18)
Other Considerations	400	(11)
Rational Functions of the Second Degree	400	(5)
The Modulus of a Quadrilateral	405	(3)
Computational Issues	408	(3)
Asymptotic Evaluation of Integrals	411	(103)
Introduction	411	(11)
Fundamental Concepts	412	(6)
Elementary Examples	418	(4)
Laplace Type Integrals	422	(17)
Integration by Parts	423	(3)
Watson's Lemma	426	(4)
Laplace's Method	430	(9)
Fourier Type Integrals	439	(9)
Integration by Parts	440	(1)
Analog of Watson's Lemma	441	(2)
The Stationary Phase Method	443	(5)
The Method of Steepest Descent	448	(26)
Laplace's Method for Complex Contours	453	(21)
Applications	474	(14)
The Stokes Phenomenon	488	(12)
Smoothing of Stokes Discontinuities	494	(6)
Related Techniques	500	(14)
WKB Method	500	(4)
The Mellin Transform Method	504	(10)
Riemann--Hilbert Problems	514	(113)
Introduction	514	(4)
Cauchy Type Integrals	518	(9)
Scalar Riemann--Hilbert Problems	527	(19)
Closed Contours	529	(4)
Open Contours	533	(5)
Singular Integral Equations	538	(8)
Applications of Scalar Riemann--Hilbert Problems	546	(33)
Riemann--Hilbert Problems on the Real Axis	558	(8)
The Fourier Transform	566	(1)
The Radon Transform	567	(12)
Matrix Riemann--Hilbert Problems	579	(19)
The Riemann--Hilbert Problem for Rational Matrices	584	(2)
Inhomogeneous Riemann--Hilbert Problems and Singular Equations	586	(1)
The Riemann--Hilbert Problem for Triangular Matrices	587	(2)
Some Results on Zero Indices	589	(9)
The DBAR Problem	598	(6)
Generalized Analytic Functions	601	(3)
Applications of Matrix Riemann--Hilbert Problems and ∂ Problems	604	(23)
Appendix A Answers to Odd-Numbered Exercises	627	(10)
Bibliography	637	(3)
Index	640