Preface
1 Complex Numbers
1.1 The Algebra of Complex Numbers
1.2 Point Representation of Complex Numbers
1.3 Vectors and Polar Forms
1.4 The Complex Exponential
1.5 Powers and Roots
1.6 Planar Sets
1.7 The Riemann Sphere and Stereographic Projection
Summary
2 Analytic Functions
2.1 Functions of a Complex Variable
2.2 Limits and Continuity
2.3 Analyticity
2.4 The Cauchy-Riemann Equations
2.5 Harmonic Functions
2.6 *Steady-State Temperature as a Harmonic Function
2.7 *Iterated Maps:Julia and Mandelbrot Sets
Summary
3 Elementary Eunctions
3.1 Polynomials and Rational Functions
3.2 The Exponential,Trigonometric,and Hyperbolic Functions
3.3 The Logarithmic Function
3.4 Washers,Wedges,and Walls
3.5 Complex Powers and Inverse Trigonometric Functions
3.6 *Application to Oscillating Systems
Summary
4 Complex Integration
4.1 Contours
4.2 Contour Integrals
4.3 Independence of Path
4.4 Cauchy's Integral Theorem
4.4a Deformation of Contours Approach
4.4b Vector Analysis Approach
4.5 Cauchy's Intergral Formula and Its Consequences
4.6 Bounds for Analytic Functions
4.7 *Applications to Harmonic Functions
Summary
5 Series Representations for Analytic Functions
5.1 Sequences and Series
5.2 Taylor Series
5.3 Power Series
5.4 *Mathematical Theory of Convergence
5.5 Laurent Series
5.6 Zeros and Singularities
5.7 The Point at Infinity
5.8 *Analytic Continuation
Summary
6 Residue Theory
6.1 The Residue Theorem
6.2 Trigonometric Integrals over[0,2π]
6.3 Improper Integrals of Certain Functions over
6.4 Improper Integrals Involving Trigonometric Functions
6.5 Indented Contours
6.6 Integrals Involving Multiple-Valued Functions
6.7 The Argument Principle and Rouche's Theorem
Summary
7 Conformal Mapping
7.1 Invariance of Laplace's Equation
7.2 Geometric Considerations
7.3 Mobius Transformations
7.4 Mobius Transformations,Continued
7.5 The Schwarz-Christoffel Transformation
7.6 Applications in Electrostatics,Heat Flow,and Fluid Mechanics
7.7 Further Physical Applications of Conformal Mapping
Summary
8 The Transorms of Applied Mathematics
8.1 Fourier Series(The Finite Fourier Transform)
8.2 The Fourier Transform
8.3 The Laplace Transform
8.4 The z-Transform
8.5 Cauchy Integrals and the Hilbert Transform
Summary
A Numerical Construction of Confomal Maps
A.1 The Schwarz-Christoffel Parameter Problem
A.2 Examples
A.3 Numerical Integration
A.4 Conformal Mapping of Smooth Domains
A.5 Conformal Mapping Software
B Table of Conformal Mappings
B.1 Mobius Transformations
B.2 Other Transformations
Answers to Odd-Numbered Problems
Index
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