Foundations of Complex Analysis,9781842652237

Foundations of Complex Analysis

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Edition: 2nd
ISBN13: 9781842652237
ISBN10: 1842652230
Format: Hardcover
Pub. Date: 2005-01-30
Publisher(s): Alpha Science International
Availability: This title is currently not available.
List Price: $70.40

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Summary

"Foundations of Complex Analysis is aimed at giving students a good foundation of complex analysis and provides a basis for solving problems in mathematics, physics, engineering and many other sciences. Each chapter is supplemented with well-structured examples, and exercises with hints and outlines for solutions. This book can be used as a textbook for a two semester course in complex analysis, or as a supplementary text for an advanced course in function theory."--BOOK JACKET.

Table of Contents

Preface to the Second Edition vii
Complex Numbers
1(32)
Definition of Complex Numbers
1(4)
Geometric Interpretation
5(7)
Square roots
12(1)
Rational Powers of a Complex Number
13(4)
Topology of the Complex Plane
17(6)
Sequences and Series
23(7)
Exercises
30(3)
Functions, Limit and Continuity
33(30)
One-to-one and Onto Functions
33(4)
Concepts of Limit and Continuity
37(8)
Stereographic Projection
45(10)
Sequences and Series of Functions
55(5)
Exercises
60(3)
Analytic Functions and Power Series
63(54)
Differentiability and Cauchy-Riemann Equations
63(13)
Harmonic Functions
76(10)
Power-Series as an Analytic Function
86(9)
Exponential and Trigonometric Functions
95(6)
Logarithmic Functions
101(9)
Inverse Functions
110(1)
Exercises
111(6)
Complex Integration
117(74)
Curves in the Complex Plane
117(4)
Properties of Complex Line Integrals
121(13)
Cauchy-Goursat Theorem
134(8)
Consequence of Simply Connectivity
142(1)
Winding Number or Index of a Curve
143(3)
Homotopy Version of Cauchy's Theorem
146(5)
Cauchy Integral Formula
151(11)
Morera's Theorem
162(2)
Existence of Harmonic Conjugate
164(2)
Taylor's Theorem
166(4)
Zeros of Analytic Functions
170(7)
Laurent Series
177(9)
Exercises
186(5)
Conformal Mappings and Mobius Transformations
191(40)
Principle of Conformal Mapping
191(9)
Basic Properties of Mobius Maps
200(6)
Fixed Points and Mobius Maps
206(3)
Triples to Triples under Mobius Maps
209(2)
The Cross-Ratio and its Invariance Property
211(2)
Conformal Self-maps of Disks and Half-planes
213(10)
Principle of Symmetry and Mobius Maps
223(4)
Exercises
227(4)
Maximum Principle, Schwarz' Lemma, and Liouville's Theorem
231(32)
Maximum Modulus Principle
231(5)
Hadamard's Three Circles/Lines Theorems
236(3)
Schwarz' Lemma and its Consequences
239(10)
Liouville's Theorem
249(6)
Doubly Periodic Entire Functions
255(1)
Fundamental Theorem of Algebra
256(2)
Zeros of certain Polynomials
258(2)
Exercises
260(3)
Classification of Singularities
263(24)
Isolated and Non-isolated Singularities
263(3)
Removable Singularities
266(3)
Poles
269(2)
Further Illustrations through Laurent's Series
271(3)
Isolated Singularities at Infinity
274(3)
Meromorphic Functions
277(2)
Essential Singularities and Picard's Theorem
279(5)
Exercises
284(3)
Calculus of Residues
287(28)
Residue at a Finite Point
287(10)
Residue at the Point at Infinity
297(2)
Residue Theorem
299(4)
Number of Zeros and Poles
303(5)
Rouche's Theorem
308(3)
Exercises
311(4)
Evaluation of certain Integrals
315(32)
Integrals of Type ∫2π+α R(cos θ, sin θ) dθ
315(5)
Integrals of Type ∫α∞ -∞ f(x) dx
320(9)
Integrals of Type ∫∞ -∞ g(x) cos mx dx
329(2)
Singularities on the Real Axis
331(6)
Integrals Involving Branch Points
337(3)
Estimation of Sums
340(5)
Exercises
345(2)
Analytic Continuation
347(26)
Direct Analytic Continuation
347(8)
Monodromy Theorem
355(5)
Poisson Integral Formula
360(8)
Analytic Continuation via Reflection
368(3)
Exercises
371(2)
Representations for Meromorphic and Entire Functions
373(56)
Infinite Sums and Meromorphic Functions
374(6)
Infinite Product of Complex Numbers
380(6)
Infinite Products of Analytic Functions
386(4)
Factorization of Entire Functions
390(10)
The Gamma Function
400(3)
The Zeta Function
403(6)
Jensen's Formula
409(6)
The Order and the Genus of Entire Functions
415(10)
Exercises
425(4)
Mapping Theorems
429(36)
Open Mapping Theorem and Hurwitz' Theorem
429(3)
Basic Results on Univalent Functions
432(4)
Normal Families
436(5)
The Riemann Mapping Theorem
441(8)
Bieberbach Conjecture
449(3)
The Bloch-Landau Theorems
452(6)
Picard's Theorem
458(4)
Exercises
462(3)
Bibliography 465(2)
Index of Special Notations 467(4)
Hints and Solutions for Selected Exercises 471(32)
Index 503

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