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Complex Analysis and Conformal Mapping

Complex Analysis and Conformal Mappingby Peter J. OlverUniversity of MinnesotaContents1. Introduction .. 22. Complex Functions .. 2 Examples of Complex Functions .. 53. Complex Differentiation .. 9 Power Series and Analyticity .. 124. Harmonic Functions .. 15 Applications to Fluid Mechanics .. 205. Conformal Mapping .. 27 Analytic Maps .. 27 Conformality .. 33 Composition and the Riemann Mapping Theorem .. 37 Annular Domains .. 416. Applications of Conformal Mapping .. 43 Applications to Harmonic Functions and Laplace s Equation.. 43 Applications to Fluid Flow .. 48 Poisson s Equation and the Green s Function ..537. Complex Integration .. 56 Cauchy s Theorem .. 60 Circulation and Lift .. 65 Cauchy s Integral Formula.

For example, the monomial function f(z) = z3 can be expanded and written as z3 = (x+ iy)3 = (x3 − 3xy2)+ i(3x2y−y3), and so Re z3 = x3 −3xy2, Imz3 = 3x2y−y3. Many of the well-known functions appearing in real-variable calculus — polynomials, rational functions, exponentials, trigonometric functions, logarithms, and many more —

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