An Introduction to Complex Analysis

✍ Scribed by Ravi P. Agarwal, Kanishka Perera, Sandra Pinelas (auth.)

• Publisher: Springer US
• Year: 2011
• Tongue: English
• Leaves: 346
• Edition: 1
• Category: Library

✦ Synopsis

This textbook introduces the subject of complex analysis to advanced undergraduate and graduate students in a clear and concise manner.

Key features of this textbook:

-Effectively organizes the subject into easily manageable sections in the form of 50 class-tested lectures

- Uses detailed examples to drive the presentation

-Includes numerous exercise sets that encourage pursuing extensions of the material, each with an “Answers or Hints” section

-covers an array of advanced topics which allow for flexibility in developing the subject beyond the basics

-Provides a concise history of complex numbers

An Introduction to Complex Analysis will be valuable to students in mathematics, engineering and other applied sciences. Prerequisites include a course in calculus.

✦ Table of Contents

Front Matter....Pages i-xiv
Complex Numbers I....Pages 1-5
Complex Numbers II....Pages 6-10
Complex Numbers III....Pages 11-19
Set Theory in the Complex Plane....Pages 20-27
Complex Functions....Pages 28-36
Analytic Functions I....Pages 37-41
Analytic Functions II....Pages 42-51
Elementary Functions I....Pages 52-56
Elementary Functions II....Pages 57-63
Mappings by Functions I....Pages 64-68
Mappings by Functions II....Pages 69-76
Curves, Contours, and Simply Connected Domains....Pages 77-82
Complex Integration....Pages 83-90
Independence of Path....Pages 91-95
Cauchy-Goursat Theorem....Pages 96-101
Deformation Theorem....Pages 102-110
Cauchy’s Integral Formula....Pages 111-115
Cauchy’s Integral Formula for Derivatives....Pages 116-124
The Fundamental Theorem of Algebra....Pages 125-131
Maximum Modulus Principle....Pages 132-137
Sequences and Series of Numbers....Pages 138-144
Sequences and Series of Functions....Pages 145-150
Power Series....Pages 151-158
Taylor’s Series....Pages 159-168
Laurent’s Series....Pages 169-176
Zeros of Analytic Functions....Pages 177-182
Analytic Continuation....Pages 183-189
Symmetry and Reflection....Pages 190-194
Singularities and Poles I....Pages 195-199
Singularities and Poles II....Pages 200-206
Cauchy’s Residue Theorem....Pages 207-214
Evaluation of Real Integrals by Contour Integration I....Pages 215-219
Evaluation of Real Integrals by Contour Integration II....Pages 220-228
Indented Contour Integrals....Pages 229-234
Contour Integrals Involving Multi-valued Functions....Pages 235-241
Summation of Series....Pages 242-246
Argument Principle and Rouché and Hurwitz Theorems....Pages 247-252
Behavior of Analytic Mappings....Pages 253-257
Conformal Mappings....Pages 258-266
Harmonic Functions....Pages 267-274
The Schwarz-Christoffel Transformation....Pages 275-280
Infinite Products....Pages 281-286
Weierstrass’s Factorization Theorem....Pages 287-292
Mittag-Leffler Theorem....Pages 293-297
Periodic Functions....Pages 298-302
The Riemann Zeta Function....Pages 303-307
Bieberbach’s Conjecture....Pages 308-311
The Riemann Surfaces....Pages 312-315
Julia and Mandelbrot Sets....Pages 316-320
History of Complex Numbers....Pages 321-325
Back Matter....Pages 327-331

✦ Subjects

Functions of a Complex Variable; Analysis