Course Outline
Complex Analysis
COMPLEX ANALYSIS
Analytic functions
- Complex differentiation and the Cauchy-Riemann equations. Examples. Conformal mappings. Informal discussion of branch points, examples of log z and zc. [3]
Contour integration and Cauchy’s theorem
- Contour integration (for piecewise continuously differentiable curves). Statement and proof of Cauchy’s theorem for star domains. Cauchy’s integral formula, maximum modulus theorem, Liouville’s theorem, fundamental theorem of algebra. Morera’s theorem. [5]
Expansions and singularities
- Uniform convergence of analytic functions; local uniform convergence. Differentiability of a power series. Taylor and Laurent expansions. Principle of isolated zeros. Residue at an isolated singularity. Classification of isolated singularities. [4]
The residue theorem
- Winding numbers. Residue theorem. Jordan’s lemma. Evaluation of definite integrals by contour integration. Rouch´e’s theorem, principle of the argument. Open mapping theorem. [4]
Appropriate books
- L.V. Ahlfors Complex Analysis. McGraw–Hill 1978
- † A.F. Beardon Complex Analysis. Wiley
- D.J.H.Garling A Course in Mathematical Analysis (Vol 3). Cambridge University Press 2014
- † H.A. Priestley Introduction to Complex Analysis. Oxford University Press 2003
- I. Stewart and D. Tall Complex Analysis. Cambridge University Press 1983
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