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Main Question or Discussion Point
- Author: James Brown, Ruel Churchill
- Title: Complex Variables and Applications
- Amazon link https://www.amazon.com/dp/0073051942/?tag=pfamazon01-20
Table of Contents:
Code:
[LIST]
[*] Preface
[*] Complex Numbers
[LIST]
[*] Sums and Products
[*] Basic Algebraic Properties
[*] Further Properties
[*] Vectors and Moduli
[*] Complex Conjugates
[*] Exponential Form
[*] Products and Powers in Exponential Form
[*] Arguments of Products and Quotients
[*] Roots of Complex Numbers
[*] Examples
[*] Regions in the Complex Plane
[/LIST]
[*] Analytic Functions
[LIST]
[*] 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
[/LIST]
[*] Elementary Functions
[LIST]
[*] 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
[/LIST]
[*] Integrals
[LIST]
[*] Derivatives of Functions w(t)
[*] Definite Integrals of Functions w(t)
[*] Contours
[*] Contour Integrals
[*] Some Examples
[*] Examples with Branch Cuts
[*] Upper Bounds for Moduli of Contour Integrals
[*] Antiderivatives
[*] Proof of the Theorem
[*] Cauchy-Goursat Theorem
[*] Proof of the Theorem
[*] Simply Connected Domains
[*] Multiply Connected Domains
[*] Cauchy Integral Formula
[*] An Extension of the Cauchy Integral Formula
[*] Some Consequences of the Extension
[*] Liouville's Theorem and the Fundamental Theorem of Algebra
[*] Maximum Modulus Principle
[/LIST]
[*] Series
[LIST]
[*] Convergence of Sequences
[*] Convergence of Series
[*] Taylor Series
[*] Proof of Taylor's Theorem
[*] Examples
[*] Laurent Series
[*] Proof of Laurent's Theorem
[*] 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
[/LIST]
[*] Residues and Poles
[LIST]
[*] Isolated Singular Points
[*] Residues
[*] Cauchy's Residue Theorem
[*] Residue at Infinity
[*] The Three Types of Isolated Singular Points
[*] Residues at Poles
[*] Examples
[*] Zeros of Analytic Functions
[*] Zeros and Poles
[*] Behavior of Functions Near Isolated Singular Points
[/LIST]
[*] Applications of Residues
[LIST]
[*] 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
[*] Rouché's Theorem
[*] Inverse Laplace Transforms
[*] Examples
[/LIST]
[*] Mapping by Elementary Functions
[LIST]
[*] Linear Transformations
[*] The Transformation w = 1/z
[*] Mappings by 1/z
[*] Linear Fractional Transformations
[*] An Implicit Form
[*] Mappings of the Upper Half Plane
[*] The Transformation w = sin z
[*] Mappings by z^2 and Branches of z^{1/2}
[*] Square Roots of Polynomials
[*] Riemann Surfaces
[*] Surfaces for Related Functions
[/LIST]
[*] Conformal Mapping
[LIST]
[*] Preservation of Angles
[*] Scale Factors
[*] Local Inverses
[*] Harmonic Conjugates
[*] Transformations of Harmonic Functions
[*] Transformations of Boundary Conditions
[/LIST]
[*] Applications of Conformal Mapping
[LIST]
[*] 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
[/LIST]
[*] The Schwarz--Christoffel Transformation
[LIST]
[*] 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
[/LIST]
[*] Integral Formulas of the Poisson Type
[LIST]
[*] Poisson Integral Formula
[*] Dirichlet Problem for a Disk
[*] Related Boundary Value Problems
[*] Schwarz Integral Formula
[*] Dirichlet Problem for a Half Plane
[*] Neumann Problems
[/LIST]
[*] Appendixes
[LIST]
[*] Bibliography
[*] Table of Transformations of Regions
[/LIST]
[*] Index
[/LIST]
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