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Complex Varialbes and Applications (Eight Edition) James Ward Brown and Ruel V. Churchill



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Complex Varialbes and Applications (Eight 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 that are prominent in applications of the subject. The second objective is to furnish an introduction to applications of residues and conformal mapping. With regard to residues, special emphasis is given to their use in evaluating real improper integrals, finding inverse Laplace transforms, and locating zeros of functions. As for conformal mapping, considerable attention is paid to its use in solving boundary value problems that arise in studies of heat conduction and fluid flow. Hence the book may be considered as a companion volume to the authors’ text “Fourier Series and Boundary Value Problems,” where another classical method for solving boundary value problems in partial differential equations is developed. The first nine chapters of this book have for many years formed the basis of a three-hour course given each term at The University of Michigan. The classes have consisted mainly of seniors and graduate students concentrating in mathematics, engineering, or one of the physical sciences. Before taking the course, the students have completed at least a three-term calculus sequence and a first course in ordinary differential equations. Much of the material in the book need not be covered in the lectures and can be left for self-study or used for reference. If mapping by elementary functions is desired earlier in the course, one can skip to Chap. 8 immediately after Chap. 3 on elementary functions. In order to accommodate as wide a range of readers as possible, there are footnotes referring to other texts that give proofs and discussions of the more delicate results from calculus and advanced calculus that are occasionally needed. A bibliography of other books on complex variables, many of which are more advanced, is provided in Appendix 1. A table of conformal transformations that are useful in applications appears in Appendix 2.

Complex Varialbes and Applications (Eight 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
    Convergence of Series
    Taylor Series
    Examples
    Laurent Series
    Examples
    Absolute and Uniform Convergence of Power Series
    Continuity of Sums of Power Series
    Integration and Differentiation of Power Series
    Uniqueness of Series Representations
    Multiplication and Division of Power Series

  • 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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