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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.

rational functions, exponentials, trigonometric functions, logarithms, and many more — have natural complex extensions. For example, complex polynomials p(z) = anzn+ a n−1 z n−1 + ···+ a 1 z+a0 (2.2) are complex linearcombinations (meaning thatthe coefficients akareallowed tobe complex numbers) of the basic monomial functions zk= (x+ ...

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