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

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Murray R. Spiegel, Seymour Lipschutz, John Schiller and Dennis Spellman: Complex Variables (2nd Edition)

Published $\text {2009}$, Schaum

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