Book:Murray R. Spiegel/Complex Variables/Second Edition

Subject Matter

 * Complex Analysis

Contents

 * Chapter 1 Complex Numbers
 * 1.1 The Real Number System
 * 1.2 Graphical Representation of Real Numbers
 * 1.3 The Complex Number System
 * 1.4 Fundamental Operations with Complex Numbers
 * 1.5 Absolute Value
 * 1.6 Axiomatic Foundation of the Complex Number System
 * 1.7 Graphical Representation of Complex Numbers
 * 1.8 Polar Form of Complex Numbers
 * 1.9 De Moivre’s Theorem
 * 1.10 Roots of Complex Numbers
 * 1.11 Euler’s Formula
 * 1.12 Polynomial Equations
 * 1.13 The nth Roots of Unity
 * 1.14 Vector Interpretation of Complex Numbers
 * 1.15 Stereographic Projection
 * 1.16 Dot and Cross Product
 * 1.17 Complex Conjugate Coordinates
 * 1.18 Point Sets


 * Chapter 2 Functions, Limits and Continuity
 * 2.1 Variables and Functions
 * 2.2 Single and Multiple-Valued Functions
 * 2.3 Inverse Functions
 * 2.4 Transformations
 * 2.5 Curvilinear Coordinates
 * 2.6 The Elementary Functions
 * 2.7 Branch Points and Branch Lines
 * 2.8 Riemann Surfaces
 * 2.9 Limits
 * 2.10 Theorems on Limits
 * 2.11 Infinity
 * 2.12 Continuity
 * 2.13 Theorems on Continuity
 * 2.14 Uniform Continuity
 * 2.15 Sequences
 * 2.16 Limit of a Sequence
 * 2.17 Theorems on Limits of Sequences
 * 2.18 Infinite Series


 * Chapter 3 Complex Differentiation and the Cauchy-Riemann Equations
 * 3.1 Derivatives
 * 3.2 Analytic Functions
 * 3.3 Cauchy–Riemann Equations
 * 3.4 Harmonic Functions
 * 3.5 Geometric Interpretation of the Derivative
 * 3.6 Differentials
 * 3.7 Rules for Differentiation
 * 3.8 Derivatives of Elementary Functions
 * 3.9 Higher Order Derivatives
 * 3.10 L’Hospital’s Rule
 * 3.11 Singular Points
 * 3.12 Orthogonal Families
 * 3.13 Curves
 * 3.14 Applications to Geometry and Mechanics
 * 3.15 Complex Differential Operators
 * 3.16 Gradient, Divergence, Curl, and Laplacian


 * Chapter 4 Complex Integration and Cauchy's Theorem
 * 4.1 Complex Line Integrals
 * 4.2 Real Line Integrals
 * 4.3 Connection Between Real and Complex Line Integrals
 * 4.4 Properties of Integrals
 * 4.5 Change of Variables
 * 4.6 Simply and Multiply Connected Regions
 * 4.7 Jordan Curve Theorem
 * 4.8 Convention Regarding Traversal of a Closed Path
 * 4.9 Green’s Theorem in the Plane
 * 4.10 Complex Form of Green’s Theorem
 * 4.11 Cauchy’s Theorem. The Cauchy–Goursat Theorem
 * 4.12 Morera’s Theorem
 * 4.13 Indefinite Integrals
 * 4.14 Integrals of Special Functions
 * 4.15 Some Consequences of Cauchy’s Theorem


 * Chapter 5 Cauchy's Integral Formulas and Related Theorems
 * 5.1 Cauchy's Integral Formulas
 * 5.2 Some Important Theorems


 * Chapter 6 Infinite Series: Taylor's and Laurent's Series
 * 6.1 Sequences of Functions
 * 6.2 Series of Functions
 * 6.3 Absolute Convergence
 * 6.4 Uniform Convergence of Sequences and Series
 * 6.5 Power Series
 * 6.6 Some Important Theorems
 * 6.7 Taylor’s Theorem
 * 6.8 Some Special Series
 * 6.9 Laurent’s Theorem
 * 6.10 Classification of Singularities
 * 6.11 Entire Functions
 * 6.12 Meromorphic Functions
 * 6.13 Lagrange’s Expansion
 * 6.14 Analytic Continuation


 * Chapter 7 The Residue Theorem: Evaluation of Integrals and Series
 * 7.1 Residues
 * 7.2 Calculation of Residues
 * 7.3 The Residue Theorem
 * 7.4 Evaluation of Definite Integrals
 * 7.5 Special Theorems Used in Evaluating Integrals
 * 7.6 The Cauchy Principal Value of Integrals
 * 7.7 Differentiation Under the Integral Sign. Leibnitz’s Rule
 * 7.8 Summation of Series
 * 7.9 Mittag–Leffler’s Expansion Theorem
 * 7.10 Some Special Expansions


 * Chapter 8 Conformal Mapping
 * 8.1 Transformations or Mappings
 * 8.2 Jacobian of a Transformation
 * 8.3 Complex Mapping Functions
 * 8.4 Conformal Mapping
 * 8.5 Riemann’s Mapping Theorem
 * 8.6 Fixed or Invariant Points of a Transformation
 * 8.7 Some General Transformations
 * 8.8 Successive Transformations
 * 8.9 The Linear Transformation
 * 8.10 The Bilinear or Fractional Transformation
 * 8.11 Mapping of a Half Plane onto a Circle
 * 8.12 The Schwarz–Christoffel Transformation
 * 8.13 Transformations of Boundaries in Parametric Form
 * 8.14 Some Special Mappings


 * Chapter 9 Physical Applications of Conformal Mapping
 * 9.1 Boundary Value Problems
 * 9.2 Harmonic and Conjugate Functions
 * 9.3 Dirichlet and Neumann Problems
 * 9.4 The Dirichlet Problem for the Unit Circle. Poisson’s Formula
 * 9.5 The Dirichlet Problem for the Half Plane
 * 9.6 Solutions to Dirichlet and Neumann Problems by Conformal Mapping Applications to Fluid Flow
 * 9.7 Basic Assumptions
 * 9.8 The Complex Potential
 * 9.9 Equipotential Lines and Streamlines
 * 9.10 Sources and Sinks
 * 9.11 Some Special Flows
 * 9.12 Flow Around Obstacles
 * 9.13 Bernoulli’s Theorem
 * 9.14 Theorems of Blasius Applications to Electrostatics
 * 9.15 Coulomb’s Law
 * 9.16 Electric Field Intensity. Electrostatic Potential
 * 9.17 Gauss’ Theorem
 * 9.18 The Complex Electrostatic Potential
 * 9.19 Line Charges
 * 9.20 Conductors
 * 9.21 Capacitance Applications to Heat Flow
 * 9.22 Heat Flux
 * 9.23 The Complex Temperature


 * Chapter 10 Special Topics
 * 10.1 Analytic Continuation
 * 10.2 Schwarz’s Reflection Principle
 * 10.3 Infinite Products
 * 10.4 Absolute, Conditional and Uniform Convergence of Infinite Products
 * 10.5 Some Important Theorems on Infinite Products
 * 10.6 Weierstrass’ Theorem for Infinite Products
 * 10.7 Some Special Infinite Products
 * 10.8 The Gamma Function
 * 10.9 Properties of the Gamma Function
 * 10.10 The Beta Function
 * 10.11 Differential Equations
 * 10.12 Solution of Differential Equations by Contour Integrals
 * 10.13 Bessel Functions
 * 10.14 Legendre Functions
 * 10.15 The Hypergeometric Function
 * 10.16 The Zeta Function
 * 10.17 Asymptotic Series
 * 10.18 The Method of Steepest Descents
 * 10.19 Special Asymptotic Expansions
 * 10.20 Elliptic Functions