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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  only

Text: "Basic Complex Analysis", 3rd Edition, by Jerrold E. Marsden and Michael J. Hoffman; Publisher: W.H. Freeman, 1999; ISBN: 978-0-716728771

AMS 503 Instructor page

Learning Outcomes:

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.