Complex Analysis (Google eBook)

Front Cover
Springer Science & Business Media, May 18, 2001 - Mathematics - 478 pages
4 Reviews
The book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. The book consists of three parts. The first part comprises the basic core of a course in complex analysis for junior and senior undergraduates. The second part includes various more specialized topics as the argument principle, the Schwarz lemma and hyperbolic geometry, the Poisson integral, and the Riemann mapping theorem. The third part consists of a selection of topics designed to complete the coverage of all background necessary for passing PhD qualifying exams in complex analysis. Topics selected 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. The book is based on lectures given over the years by the author at several places, particularly the Interuniversity Summer School at Perugia (Italy), and also UCLA, Brown University, Valencia (Spain), and La Plata (Argentina). A native of Minnesota, the author did his undergraduate work at Yale University and his graduate work at UC Berkeley. After spending some time at MIT and at the Universidad Nacional de La Plata (Argentina), he joined the faculty at UCLA in 1968. The author has published a number of research articles and several books on functional analysis and analytic function theory. he is currently involved in the California K-12 education scene.
  

What people are saying - Write a review

User Review - Flag as inappropriate

1

User Review - Flag as inappropriate

A beautiful book.

Contents

The Complex Plane and Elementary Functions
1
2 Polar Representation
5
3 Stereographic Projection
11
4 The Square and Square Root Functions
15
5 The Exponential Function
19
6 The Logarithm Function
21
7 Power Functions and Phase Factors
24
8 Trigonometric and Hyperbolic Functions
29
2 Rouches Theorem
229
3 Hurwitzs Theorem
231
4 Open Mapping and Inverse Function Theorems
232
5 Critical Points
236
6 Winding Numbers
242
7 The Jump Theorem for Cauchy Integrals
246
8 Simply Connected Domains
252
The Schwarz Lemma and Hyperbolic Geometry
260

Analytic Functions
33
2 Analytic Functions
42
3 The CauchyRiemann Equations
46
4 Inverse Mappings and the Jacobian
51
5 Harmonic Functions
54
6 Conformal Mappings
58
7 Fractional Linear Transformations
63
Line Integrals and Harmonic Functions
70
2 Independence of Path
76
3 Harmonic Conjugates
83
4 The Mean Value Property
85
5 The Maximum Principle
87
6 Applications to Fluid Dynamics
90
7 Other Applications to Physics
97
Complex Integration and Analyticity
102
2 Fundamental Theorem of Calculus for Analytic Functions
107
3 Cauchys Theorem
110
4 The Cauchy Integral Formula
113
5 Liouvilles Theorem
117
6 Moreras Theorem
119
7 Goursats Theorem
123
8 Complex Notation and Pompeius Formula
124
Power Series
130
2 Sequences and Series of Functions
133
3 Power Series
138
4 Power Series Expansion of an Analytic Function
144
5 Power Series Expansion at Infinity
149
6 Manipulation of Power Series
151
7 The Zeros of an Analytic Function
154
8 Analytic Continuation
158
Laurent Series and Isolated Singularities
165
2 Isolated Singularities of an Analytic Function
171
3 Isolated Singularity at Infinity
178
4 Partial Fractions Decomposition
179
5 Periodic Functions
182
6 Fourier Series
186
The Residue Calculus
195
2 Integrals Featuring Rational Functions
199
3 Integrals of Trigonometric Functions
203
4 Integrands with Branch Points
206
5 Fractional Residues
209
6 Principal Values
212
7 Jordans Lemma
216
8 Exterior Domains
219
The Logarithmic Integral
224
2 Conformal SelfMaps of the Unit Disk
263
3 Hyperbolic Geometry
266
Harmonic Functions and the Reflection Principle
274
2 Characterization of Harmonic Functions
280
3 The Schwarz Reflection Principle
282
Conformal Mapping
289
2 The Riemann Mapping Theorem
294
3 The SchwarzChristoffel Formula
296
4 Return to Fluid Dynamics
304
5 Compactness of Families of Functions
306
6 Proof of the Riemann Mapping Theorem
311
Compact Families of Meromorphic Functions
315
2 Theorems of Montel and Picard
320
3 Julia Sets
324
4 Connectedness of Julia Sets
333
5 The Mandelbrot Set
338
Approximation Theorems
342
2 The MittagLeffler Theorem
348
3 Infinite Products
352
4 The Weierstrass Product Theorem
358
Some Special Functions
361
2 Laplace Transforms
365
3 The Zeta Function
370
4 Dirichlet Series
376
5 The Prime Number Theorem
382
The Dirichlet Problem
390
2 Subharmonic Functions
394
3 Compactness of Families of Harmonic Functions
398
4 The Perron Method
402
5 The Riemann Mapping Theorem Revisited
406
6 Greens Function for Domains with Analytic Boundary
407
7 Greens Function for General Domains
413
Riemann Surfaces
418
2 Harmonic Functions on a Riemann Surface
426
3 Greens Function of a Surface
429
4 Symmetry of Greens Function
434
5 Bipolar Greens Function
436
6 The Uniformization Theorem
438
7 Covering Surfaces
441
Hints and Solutions for Selected Exercises
447
References
469
List of Symbols
471
Index
473
Copyright

Common terms and phrases

Popular passages

Page 1 - By a complex number we mean a number of the form z = x + iy where x and y are real numbers and i...
Page 1 - The component x is called the real part of z, and y is the imaginary part of z.

References to this book

All Book Search results »

Bibliographic information