Complex Variables (Schaum) – Murray R. Spiegel, Lipschutz, Schiller & Spellman – 2nd Edition

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
1.1 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 41

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
2.1 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 7

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
3.1 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 1

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
4.1 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 vii

CHAPTER 5 CAUCHY’S INTEGRAL FORMULAS AND RELATED THEOREMS 144
5.1 Cauchy’s Integral Formulas
5.2 Some Important Theorems

CHAPTER 6 INFINITE SERIES TAYLOR’S AND LAURENT’S SERIES 169

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
6.1 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 205

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 242

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
8.1 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 280

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
9.1 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
9.2 Heat Flux
9.23 The Complex Temperature