MA708 Complex Analysis

Course Learning Objectives: The objective of this course is to 1. derive properties of analytic functions and branch of logarithm. 2. examine properties of Mobius transformation and express analytic functions as power series. 3. derive various forms of Cauchy’s integral formula/theorems. 4. classify singularities and obtain Laurent’s series expansions. 5. discuss Cauchy’s residue theorem to evaluate integrals and identify zeros of analytic functions. 

Course Content.  Lines and planes in complex plane – Extended complex plane – Spherical representation – Power series – Sequence of functions – Differentiability – Cauchy-Riemann equations – Analytic functions and branch of logarithm. 

Mobius transformations – Cross ratios, symmetry and orientation principles – Complex integration and power series representation of analytic functions. 

Zeros of analytic functions – Liouville’s theorem – Fundamental theorem of algebra – Lucas theorem -  Identity theorem and maximum modulus theorem – Index of a closed curve – Cauchy’s theorem and integral formula on open subsets of ℂ – Homotopic version of Cauchy’s theorem – Simple connectedness, counting of zeros – Open mapping theorem and Goursat’s theorem. 

Singularities – Removable singularities and poles – Laurent series and essential singularities – CasoratiWeiestrass theorem.

 Cauchy’s residue theorem – Evaluation of real integrals – Argument principle – Rouche’s theorem -  Maximum/minimum modulus theorems and Schwarz lemma. 

Reference Books.

 1. J. B. Conway, Functions of One Complex Variables – I, 2nd edition, Narosa, 2002. 

2. D. Ullrich, Complex Made Simple, Volume 97, American Mathematical Society, 2008. 

3. H. L. Shin and E. Bernard, Classical Complex Analysis, Jones and Bartlett, 2011. 

4. L. V. Ahlfors, Complex Analysis, 3rd edition, McGraw Hill Co., 2017.

 5. W. Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill, 2017. 

Course Outcomes: On completion of this course, students will be able to 

1. analyze properties of analytic functions and construct branch of logarithm. 

2. determine properties of Mobius transformation and express analytic functions as power series. 

3. prove various forms of Cauchy’s integral formula/theorems. 

4. find singularities and residues at them as well as obtain Laurent’s series expansions. 

5. evaluate real integrals using Cauchy’s residue theorem and analyze location of zeros of analytic functions. 

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