MT 3214 Complex Analysis

Lectures

Wednesday 12:00-1:00 PM

Thursday 3:00-4:00 PM

Tutorials

Tuesday 2:00-3:00 PM

Weekly homework problems will be uploaded here.

Contents

The complex plane and elementary functions. Analytic functions, Cauchy-Riemann equations, harmonic functions, conformal mappings, Mobius transformations. Line integrals, harmonic conjugates, mean value property, maximum principle. Complex line integrals, Cauchy’s theorem, Cauchy’s integral formula, Liouville’s theorem, Morera’s theorem, Goursat theorem. Power series expansion, zeros of an analytic function. Isolated singularities, Laurent series expansion, classification of singularities. Residue theorem, evaluation of integrals. The argument principle, Rouche’s theorem, Hurwitz’s theorem, open mapping and inverse function theorems, winding numbers, simply connected domains. Schwarz lemma, automorphisms of the disc, Riemann mapping theorem.

Textbook

T. W. Gamelin, Complex Analysis

References

1. L. V. Ahlfors, Complex Analysis

2. J. B. Conway, Functions of one complex variable

3. R. Remmert, Theory of complex functions

4. E. M. Stein and Rami Shakarchi, Complex Analysis

Evaluation

End-sem 30-40%

Mid-sem 30%

Quizzes 20-30%. A quiz of duration 30 minutes will be conducted on every third tutorial session. Best 3 out of all the quizzes will be considered. There will not be any makeup quizzes.

Assignment 10%.