Schaum's Outline of Complex Variables, 2ed | Schaum's Outline

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Schaum's Outline of Complex Variables, 2ed

Schaum's Outline of Complex Variables, 2ed

Authors: Murray Spiegel, Seymour Lipschutz, John Schiller and Dennis Spellman

Published: May 2009

eISBN: 9780071615709 0071615709 | ISBN: 9780071615693

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

Index