Complex Varialbes and Applications (Seventh Edition) by James Ward Brown and Ruel V. Churchill
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Complex Varialbes and Applications (Seventh Edition) written by
James Ward Brown , Professor of Mathematics, The University of Michigan--Dearborn and
Ruel V. Churchill , Late Professor of Mathematics, The University of Michigan.
The first objective of the book is to develop those parts of the theory which are prominent in applications of the subject. The second objective is to furnish an introduction to applications of residues and conformal mapping. Special emphasis is given to the use of conformal mapping in solving boundary value problems that arise in studies of heat conduction, electrostatic potential, and fluid flow. Hence the book may be considered as a companion volume to the authors' "Fourier Series and Boundary Value Problems" and Rue! V, Churchill's "Operational Mathematics," where other classical methods for solving boundary value problems in partial differential equations are developed. The latter book also contains further applications of residues in connection with Laplace transforms.
This book has been used for many years in a three-hour course given each term at The University of Michigan. The classes have consisted mainly of seniors and graduate students majoring in mathematics, engineering, or one of the physical sciences. Before taking the course, the students have completed at least a three-term calculus sequence, a first course in ordinary differential equations, and sometimes a term of advanced calculus. In order to accommodate as wide a range of readers as possible, there are footnotes referring to texts that give proofs and discussions of the more delicate results from calculus that are occasionally needed. Some of the material in the book need not be covered in lectures and can be left for students to read on their own. If mapping by elementary functions and applications of conformal mapping are desired earlier in the course, one can skip to Chapters 8, 9, and 10 immediately after Chapter 3 on elementary functions.
Complex Varialbes and Applications (Seventh Edition) written by James Ward Brown and Ruel V. Churchill
cover the following topics.
Preface
1. Complex Numbers
Sums and Products 1 Basic Algebraic Properties
Further Properties
Moduli
Complex Conjugates
Exponential Form
Products and Quotients in Exponential Form
Roots of Complex Numbers
Examples
Regions in the Complex Plane
2. Analytic Functions
Functions of a Complex Variable
Mappings
Mappings by the Exponential Function
Limits
Theorems on Limits
Limits Involving the Point at Infinity
Continuity
Derivatives
Differentiation Formulas
Cauchy-Riemann Equations
Sufficient Conditions for Differentiability
Polar Coordinates
Analytic Functions
Examples
Harmonic Functions
Uniquely Determined Analytic Functions
Reflection Principle
3. Elementary Functions
The Exponential Function
The Logarithmic Function
Branches and Derivatives of Logarithms
Some Identities Involving Logarithms
Complex Exponents
Trigonometric Functions
Hyperbolic Functions
Inverse Trigonometric and Hyperbolic Functions
4. Integrals
Derivatives of Functions w(t)
Definite Integrals of Functions w(t)
Contours
Contour Integrals
Examples
Upper Bounds for Moduli of Contour Integrals
Antiderivatives
Examples
Cauchy-Goursat Theorem
Proof of the Theorem
Simply and Multiply Connected Domains
Cauchy Integral Formula
Derivatives of Analytic Functions
Liouville's Theorem and the Fundamental Theorem of Algebra
Maximum Modulus Principle
5. Series
Convergence of Sequences 175
Convergence of Series 178
Taylor Series 182
Examples 185
Laurent Series 190
Examples 195
Absolute and Uniform Convergence of Power Series 200
Continuity of Sums of Power Series 204
Integration and Differentiation of Power Series 206
Uniqueness of Series Representations 210
Multiplication and Division of Power Series 215
6. Residues and Poles
Residues
Cauchy's Residue Theorem
Using a Single Residue
The Three Types of Isolated Singular Points
Residues at Poles
Examples
Zeros of Analytic Functions
Zeros and Poles
Behavior off Near Isolated Singular Points
7. Applications of Residues
Evaluation of Improper Integrals
Example
Improper Integrals from Fourier Analysis
Jordan's Lemma
Indented Paths
An Indentation Around a Branch Point
Integration Along a Branch Cut
Definite integrals involving Sines and Cosines
Argument Principle
Rouch6's Theorem
Inverse Laplace Transforms
Examples
8. Mapping by Elementary Functions
Linear Transformations
The Transformation w = liz
Mappings by 1/z
Linear Fractional Transformations
An Implicit Form
Mappings of the Upper Half Plane
The Transformation w = sin z
Mappings by z"' and Branches of z
Square Roots of Polynomials
Riemann Surfaces
Surfaces for Related Functions
9. Conformal Mapping
Preservation of Angles
Scale Factors
Local Inverses
Harmonic Conjugates
Transformations of Harmonic Functions
Transformations of Boundary Conditions
10. Applications of Conformal Mapping
Steady Temperatures
Steady Temperatures in a Half Plane
A Related Problem
Temperatures in a Quadrant
Electrostatic Potential
Potential in a Cylindrical Space
Two-Dimensional Fluid Flow
The Stream Function
Flows Around a Corner and Around a Cylinder
11. The Schwarz-Christoffel Transformation
Mapping the Real Axis onto a Polygon
Schwarz-Christoffel Transformation
Triangles and Rectangles
Degenerate Polygons
Fluid Flow in a Channel Through a Slit
Flow in a Channel with an Offset
Electrostatic Potential about an Edge of a Conducting Plate
12. Integral Formulas of the Poisson Type
Poisson Integral Formula
Dirichlet Problem for a Disk
Related Boundary Value Problems
Schwarz Integral Formula
Dirichiet Problem for a Half Plane
Neumann Problems
Appendixes
Bibliography
Table of Transformations of Regions
Index
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