Complex Analysis and Conformal Mapping

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Complex Analysis and Conformal Mappingby Peter J. OlverUniversity of MinnesotaContents1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Complex Functions. . . . . . . . . . . . . . . . . . . . . . . . . 2Examples of Complex Functions . . . . . . . . . . . . . . . . . . . 53. Complex Differentiation. . . . . . . . . . . . . . . . . . . . . . . 9Power Series and Analyticity . . . . . . . . . . . . . . . . . . . . 124. Harmonic Functions. . . . . . . . . . . . . . . . . . . . . . . . . 15Applications to Fluid Mechanics . . . . . . . . . . . . . . . . . . . 205. Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . 27Analytic Maps.

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 —

  Mapping, Conformal, Conformal mapping