Transcription of Complex Analysis and Conformal Mapping
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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 .. 70 Derivatives by Integration .. 72 The Calculus of Residues .. 72 Evaluation of Real Integrals.
Re 1 z Im 1 z Figure 1. Real and Imaginary Parts of f(z) = 1 z. Examples of Complex Functions (a) Harmonic Polynomials: As noted above, any complex polynomial is a linear combi-
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