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

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## Murray R. Spiegel, Seymour Lipschutz, John Schiller and Dennis Spellman:

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

Published $\text {2009}$, **Schaum**.

### Subject Matter

### 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