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.
- 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.
- Metric spaces and Topology of complex plane. Open sets in Complex plane, Few properties of metric topology, Continuity, Uniform convergence
- Elementary properties of Analytic functions. Analytic functions as mapping. Exponential and Logarithm
- 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.
- 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.
- Harmonic Functions: Basic properties, Dirichlet problem, Green function.
- Functions of One Complex Variable by John B Conway, 2nd edition, Narosa.
- Complex Analysis by Lars Ahlfors, 3rd edition, McGraw Hill Education India.
- Introduction to Complex Analysis by H A Priestley, Oxford University Press.
- Complex Function Theory by D Sarason, 2nd edition, TRIM Series, Hindustan Book Agency.
- Complex Analysis by T W Gamelin, Springer.
- Complex Variables by M J Ablowitz and A S Fokas, 2nd edition, Cambridge University Press.