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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 .. 38 Annular Domains .. 426. Applications of Conformal Mapping .. 44 Applications to Harmonic functions and Laplace s Equation.. 44 Applications to Fluid Flow .. 48 Poisson s Equation and the Green s Function ..537. Complex Integration.. 56 Cauchy s Theorem .. 61 Circulation and Lift .. 65 Cauchy s Integral Formula.

In this manner, complex functions provide a rich lode of additional solutions to the two-dimensional Laplace equation, which can be exploited in a wide range of physical and mathematical applications. One of the most useful consequences stems from the elementary observation that the composition of two complex functions is also a complex function.

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