Complex Variables and Applications / Edition 7

by James Ward Brown, Ruel V. Churchill, Ruel Vance Churchill

Overview

These classic textbooks, specializing in the techniques and applications of advanced mathematics to physical science and engineering, have endured as perennial standards for more than 60 years. The latest editions preserve the hallmark features that made Brown and Churchill a household name in advanced mathematics education-clear and concise exposition, interesting examples, and accessible level-while adding new enhancements, improved organization, and more modern examples and applications to serve another generation of students.

Complex Variables and Applications provides a one-term introduction to the theory and application of functions of a complex variable. Its primary objective is to develop those parts of the theory that are prominent in the applications of the subject. Numerous applications to the physical sciences and engineering are provided throughout, including those suitable for reference and self-study.

Fourier Series and Boundary Value Problems provides an introduction to partial differential equations for students who have completed a first course in ordinary differential equations. The text's primary objective is to develop the concepts of Fourier series and their applications to boundary value problems by finding solutions to specific problems rather than developing general theories. Detailed physical applications are provided in a straightforward and accessible manner.

Product Details

ISBN-13:	9780072872521
Publisher:	McGraw-Hill Companies, The
Publication date:	02/26/2003
Series:	International Series in Pure and Applied Mathematics
Edition description:	Older Edition
Pages:	480
Product dimensions:	6.40(w) x 9.50(h) x 0.99(d)

Table of Contents

Preface     x
Complex Numbers     1
Sums and Products     1
Basic Algebraic Properties     3
Further Properties     5
Vectors and Moduli     9
Complex Conjugates     13
Exponential Form     16
Products and Powers in Exponential Form     18
Arguments of Products and Quotients     20
Roots of Complex Numbers     24
Examples     27
Regions in the Complex Plane     31
Analytic Functions     35
Functions of a Complex Variable     35
Mappings     38
Mappings by the Exponential Function     42
Limits     45
Theorems on Limits     48
Limits Involving the Point at Infinity     50
Continuity     53
Derivatives     56
Differentiation Formulas     60
Cauchy-Riemann Equations     63
Sufficient Conditions for Differentiability     66
Polar Coordinates     68
Analytic Functions     73
Examples     75
Harmonic Functions     78
Uniquely Determined Analytic Functions     83
ReflectionPrinciple     85
Elementary Functions     89
The Exponential Function     89
The Logarithmic Function     93
Branches and Derivatives of Logarithms     95
Some Identities Involving Logarithms     98
Complex Exponents     101
Trigonometric Functions     104
Hyperbolic Functions     109
Inverse Trigonometric and Hyperbolic Functions     112
Integrals     117
Derivatives of Functions w(t)     117
Definite Integrals of Functions w(t)     119
Contours     122
Contour Integrals     127
Some Examples     129
Examples with Branch Cuts     133
Upper Bounds for Moduli of Contour Integrals     137
Antiderivatives     142
Proof of the Theorem     146
Cauchy-Goursat Theorem     150
Proof of the Theorem     152
Simply Connected Domains     156
Multiply Connected Domains     158
Cauchy Integral Formula     164
An Extension of the Cauchy Integral Formula     165
Some Consequences of the Extension     168
Liouville's Theorem and the Fundamental Theorem of Algebra      172
Maximum Modulus Principle     175
Series     181
Convergence of Sequences     181
Convergence of Series     184
Taylor Series     189
Proof of Taylor's Theorem     190
Examples     192
Laurent Series     197
Proof of Laurent's Theorem     199
Examples     202
Absolute and Uniform Convergence of Power Series     208
Continuity of Sums of Power Series     211
Integration and Differentiation of Power Series     213
Uniqueness of Series Representations     217
Multiplication and Division of Power Series     222
Residues and Poles     229
Isolated Singular Points     229
Residues     231
Cauchy's Residue Theorem     234
Residue at Infinity     237
The Three Types of Isolated Singular Points     240
Residues at Poles     244
Examples     245
Zeros of Analytic Functions     249
Zeros and Poles     252
Behavior of Functions Near Isolated Singular Points     257
Applications of Residues     261
Evaluation of Improper Integrals      261
Example     264
Improper Integrals from Fourier Analysis     269
Jordan's Lemma     272
Indented Paths     277
An Indentation Around a Branch Point     280
Integration Along a Branch Cut     283
Definite Integrals Involving Sines and Cosines     288
Argument Principle     291
Rouche's Theorem     294
Inverse Laplace Transforms     298
Examples     301
Mapping by Elementary Functions     311
Linear Transformations     311
The Transformation w = 1/z     313
Mappings by 1/z     315
Linear Fractional Transformations     319
An Implicit Form     322
Mappings of the Upper Half Plane     325
The Transformation w = sin z     330
Mappings by z[superscript 2] and Branches of z[superscript 1/2]     336
Square Roots of Polynomials     341
Riemann Surfaces     347
Surfaces for Related Functions     351
Conformal Mapping     355
Preservation of Angles     355
Scale Factors     358
Local Inverses     360
Harmonic Conjugates      363
Transformations of Harmonic Functions     365
Transformations of Boundary Conditions     367
Applications of Conformal Mapping     373
Steady Temperatures     373
Steady Temperatures in a Half Plane     375
A Related Problem     377
Temperatures in a Quadrant     379
Electrostatic Potential     385
Potential in a Cylindrical Space     386
Two-Dimensional Fluid Flow     391
The Stream Function     393
Flows Around a Corner and Around a Cylinder     395
The Schwarz-Christoffel Transformation     403
Mapping the Real Axis Onto a Polygon     403
Schwarz-Christoffel Transformation     405
Triangles and Rectangles     408
Degenerate Polygons     413
Fluid Flow in a Channel Through a Slit     417
Flow in a Channel With an Offset     420
Electrostatic Potential About an Edge of a Conducting Plate     422
Integral Formulas of the Poisson Type     429
Poisson Integral Formula     429
Dirichlet Problem for a Disk     432
Related Boundary Value Problems     437
Schwarz Integral Formula     440
Dirichlet Problem for a Half Plane     441
Neumann Problems     445
Appendixes     449
Bibliography     449
Table of Transformations of Regions     452
Index     461