MA704 Complex Analysis

The course presents an introduction to analytic functions, conformal mappings, Mobius transformations and power series. Various Cauchy’ theorems are discussed and used in evaluation of integral. It deals with locations of zeros of analytic functions and maximum principles. 

Course Content.  Lines and planes in complex plane, extended complex plane, spherical representation, power series, analytic functions as mappings, branch of logarithm, conformal mappings, Mobius transformations.

Power series representation of analytic functions, zeros of analytic functions, index of a closed curve, Cauchy’s theorem and integral formula on open subsets of C. 

Homotopy, homotopic version of Cauchy’s theorem, simple connectedness, counting of zeros, open mapping theorem, Goursat’s theorem, Classification of singularities, Laurent series.

Residue, Contour integration, argument principle, Rouche’s theorem, Maximum principle, Schwarz’ lemma.

Reference Books.

1.  Conway John. Functions of One Complex Variables. 2nd ed, Narosa, New Delhi. 2002.

2.  Ahlfors Lars. Complex Analysis. McGraw Hill Co., New York. 1988.

3. Hahn Liang-Shin and Epstein Bernard. Classical Complex Analysis. Jones and Bartlett India, New Delhi. 2011.

4.  Rudin Walter. Real and Complex Analysis. McGraw-Hill. 1987.

5.  Ullrich David. Complex Made Simple. American Math. Soc., Washington DC. 2008. 

Course Learning Outcomes. On completion of the course, student will be able to

 1.      understand analytic functions as mappings and discuss properties of conformal mappings, and Mobius transformations

2.      obtain series representation of analytic functions

3.      evaluate various integrals by using Cauchy’s residue theorem

4.      classify singularities and derive Laurent series expansion.

Here are some video lectures that one can listen to learn complex analysis:

• 1. Complex Analysis, by Steven Miller

  2. Complex Analysis   Prof. P. A. S. Sree Krishna

  3. Advanced Complex Analysis - Venkata Balaji