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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 Limis 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 Functions
3.1 Poynomials and Rational Functions
3.2 The Exponential,Trionometric,and Hyperbolic Functions
3.3 The Logarthmic Function
3.4 Washers,Wedges ,and Walls
3.5 Complex Powers and Inverse Trigonometric Functions
3.6 Application to Osclillating Systems Summary
4 Complex Integration
4.1 Contours
4.2 Contour Integrals
4.3 Independence of Path
4.4 Cauchy's Integral Theorem
4.5 Cauchy's Integral Formula and Its Consequences
4.6 Bounds for Analytic Functions
4.7 Applications to Harmonic Functions Summary
5 Series Representations for Analytic Functions
6 Residue Theory
7 Conformal Mapping
8 The Transforms of Applied Mathematics
A Numerical Construction of Conformal Maps
B Table of Conformal Mappings
Answers to Odd-Numbered Problems
Index