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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 .. 71 Derivatives by Integration .. 72 Liouville s Theorem and the Fundamental Theorem of Algebra.. 73 The Calculus of Residues .. 75 Evaluation of Real Integrals .. 80 References.. 861/7/221c 2022 Peter J. Olver1. term Complex Analysis refers to the calculus of Complex -valued functionsf(z)depending on a single Complex variablez.

and hence (2.4) does indeed define a complex-valued solution to the Laplace equation. In most applications, we are searching for real solutions, and so our complex d’Alembert-type formula (2.4) is not entirely satisfactory. As we know, a complex number z= x+ iy is real if and only if it equals its own conjugate: z= z.

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