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AMS 503, Applications of Complex Analysis
A study of those concepts and techniques in complex function theory that are of interest for their applications. Pertinent material is selected from the following topics: harmonic functions, calculus of residues, conformal mapping, and the argument principle. Application is made to problems in heat conduction, potential theory, fluid dynamics, and feedback systems.
3 credits, ABCF grading
Effective Fall 2013 Semester: This course will be offered in the Fall semester
Text: "Basic Complex Analysis", 3rd Edition, by Jerrold E. Marsden and Michael J. Hoffman; Publisher: W.H. Freeman, 1999; ISBN: 978-0-716728771
1) Demonstrate mastery of basic definitions & operations, polar form, functions, limits, compact sets, differentiation, Cauchy-Riemann equations, angles under holomorphic ("differentiable") maps.
2) Demonstrate mastery of :
* Formal & convergent power series, analytic functions, inverse & open mapping theorems, local maximum modulus principle;
* Connected sets, integrals over paths, primitives ("antiderivatives"), local Cauchy theorem;
* Winding numbers, global Cauchy Theorem.
3) Demonstrate mastery of:
* Applications of Cauchy's integral formula, Laurent series;
* Calculus of residues, evaluation of complex definite integrals, Fourier transform;
* Conformal mapping, Schwarz lemma, and applications;
* Harmonic functions;
* Schwarz reflection;
* Riemann mapping theorem;
* Analytic continuation along curves;
* Applications of Maximum Modulus Principle an Jensen's Formula.
4) Study topics on elliptic functions, Gamma & Zeta functions.