Complex Analysis

Core course for M.Sc. Mathematics

Credits: 4 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 621 (Analysis I). Undergraduates not allowed.

Overview: A graduate course of one variable complex analysis.

“The shortest path between two truths in the real domain passes through the complex domain” – Jacques Hadamard.

Detailed Syllabus:

1. The complex number system: The field of Complex numbers, the complex plane, Polar representation and roots of complex numbers, Line and Half planes in the Complex plane, the extended plane and its Stereographic representation.
2. Metric spaces and Topology of complex plane. Open sets in Complex plane, Few properties of metric topology, Continuity, Uniform convergence
3. Elementary properties of Analytic functions. Analytic functions as mapping. Exponential and Logarithm
4. Complex Integration: Basic review of Riemann-Stieltjes integral (without proof), Path integral, Power series representation of an analytic function, Liouville’s theorem and Identity theorem, Index of a closed curve, Cauchy theorem and Integral Formula, Open mapping theorem.
5. Singularities: Removable singularity and Pole, Laurent series expansion, Essential singularity and Casorati-Weierstrass theorem Residues, Solving integral, Argument Principle, Rouche’s Theorem, Maximum modulus theorem.
6. Harmonic Functions: Basic properties, Dirichlet problem, Green function.

References:

1. Functions of One Complex Variable by John B Conway, 2^nd edition, Narosa.
2. Complex Analysis by Lars Ahlfors, 3^rd edition, McGraw Hill Education India.
3. Introduction to Complex Analysis by H A Priestley, Oxford University Press.
4. Complex Function Theory by D Sarason, 2^nd edition, TRIM Series, Hindustan Book Agency.
5. Complex Analysis by T W Gamelin, Springer.
6. Complex Variables by M J Ablowitz and A S Fokas, 2^nd edition, Cambridge University Press.