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Complex Analysis : Undergraduate Texts in Mathematics - Theodore W. Gamelin

Complex Analysis

Undergraduate Texts in Mathematics

Hardcover ISBN: 9780387950938
Number Of Pages: 480

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The book provides an introduction to complex analysis for readers with some very basic familiarity with complex numbers. The sixteen chapters begin with the fundamentals at the undergraduate level, finishing with material designed to complete the coverage of all background necessary for passing PhD qualifying exams in complex analysis. Topics covered include Julia sets and the Mandelbrot set, Dirichlet series and the prime number theorem, and the uniformization theorem for Riemann surfaces. The three geometries, spherical, Euclidean, and hyperbolic, are stressed. Exercises range from the very simple to the quite challenging, in all chapters.

Prefacep. vii
Introductionp. xvii
The Complex Plane and Elementary Functionsp. 1
Complex Numbersp. 1
Polar Representationp. 5
Stereographic Projectionp. 11
The Square and Square Root Functionsp. 15
The Exponential Functionp. 19
The Logarithm Functionp. 21
Power Functions and Phase Factorsp. 24
Trigonometric and Hyperbolic Functionsp. 29
Analytic Functionsp. 33
Review of Basic Analysisp. 33
Analytic Functionsp. 42
The Cauchy-Riemann Equationsp. 46
Inverse Mappings and the Jacobianp. 51
Harmonic Functionsp. 54
Conformal Mappingsp. 58
Fractional Linear Transformationsp. 63
Line Integrals and Harmonic Functionsp. 70
Line Integrals and Green's Theoremp. 70
Independence of Pathp. 76
Harmonic Conjugatesp. 83
The Mean Value Propertyp. 85
The Maximum Principlep. 87
Applications to Fluid Dynamicsp. 90
Other Applications to Physicsp. 97
Complex Integration and Analyticityp. 102
Complex Line Integralsp. 102
Fundamental Theorem of Calculus for Analytic Functionsp. 107
Cauchy's Theoremp. 110
The Cauchy Integral Formulap. 113
Liouville's Theoremp. 117
Morera's Theoremp. 119
Goursat's Theoremp. 123
Complex Notation and Pompeiu's Formulap. 124
Power Seriesp. 130
Infinite Seriesp. 130
Sequences and Series of Functionsp. 133
Power Seriesp. 138
Power Series Expansion of an Analytic Functionp. 144
Power Series Expansion at Infinityp. 149
Manipulation of Power Seriesp. 151
The Zeros of an Analytic Functionp. 154
Analytic Continuationp. 158
Laurent Series and Isolated Singularitiesp. 165
The Laurent Decompositionp. 165
Isolated Singularities of an Analytic Functionp. 171
Isolated Singularity at Infinityp. 178
Partial Fractions Decompositionp. 179
Periodic Functionsp. 182
Fourier Seriesp. 186
The Residue Calculusp. 195
The Residue Theoremp. 195
Integrals Featuring Rational Functionsp. 199
Integrals of Trigonometric Functionsp. 203
Integrands with Branch Pointsp. 206
Fractional Residuesp. 209
Principal Valuesp. 212
Jordan's Lemmap. 216
Exterior Domainsp. 219
The Logarithmic Integralp. 224
The Argument Principlep. 224
Rouché's Theoremp. 229
Hurwitz's Theoremp. 231
Open Mapping and Inverse Function Theoremsp. 232
Critical Pointsp. 236
Winding Numbersp. 242
The Jump Theorem for Cauchy Integralsp. 246
Simply Connected Domainsp. 252
The Schwarz Lemma and Hyperbolic Geometryp. 260
The Schwarz Lemmap. 260
Conformal Self-Maps of the Unit Diskp. 263
Hyperbolic Geometryp. 266
Harmonic Functions and the Reflection Principlep. 274
The Poisson Integral Formulap. 274
Characterization of Harmonic Functionsp. 280
The Schwarz Reflection Principlep. 282
Conformal Mappingp. 289
Mappings to the Unit Disk and Upper Half-Planep. 289
The Riemann Mapping Theoremp. 294
The Schwarz-Christoffel Formulap. 296
Return to Fluid Dynamicsp. 304
Compactness of Families of Functionsp. 306
Proof of the Riemann Mapping Theoremp. 311
Compact Families of Meromorphic Functionsp. 315
Marty's Theoremp. 315
Theorems of Montel and Picardp. 320
Julia Setsp. 324
Connectedness of Julia Setsp. 333
The Mandelbrot Setp. 338
Approximation Theoremsp. 342
Runge's Theoremp. 342
The Mittag-Leffler Theoremp. 348
Infinite Productsp. 352
The Weierstrass Product Theoremp. 358
Some Special Functionsp. 361
The Gamma Functionp. 361
Laplace Transformsp. 365
The Zeta Functionp. 370
Dirichlet Seriesp. 376
The Prime Number Theoremp. 382
The Dirichlet Problemp. 390
Green's Formulaep. 390
Subharmonic Functionsp. 394
Compactness of Families of Harmonic Functionsp. 398
The Perron Methodp. 402
The Riemann Mapping Theorem Revisitedp. 406
Green's Function for Domains with Analytic Boundaryp. 407
Green's Function for General Domainsp. 413
Riemann Surfacesp. 418
Abstract Riemann Surfacesp. 418
Harmonic Functions on a Riemann Surfacep. 426
Green's Function of a Surfacep. 429
Symmetry of Green's Functionp. 434
Bipolar Green's Functionp. 436
The Uniformization Theoremp. 438
Covering Surfacesp. 441
Hints and Solutions for Selected Exercisesp. 447
Referencesp. 469
List of Symbolsp. 471
Indexp. 473
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780387950938
ISBN-10: 0387950931
Series: Undergraduate Texts in Mathematics
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 480
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.39 x 15.6  x 2.69
Weight (kg): 0.87
Edition Type: New edition

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