MAT425

Advanced Complex Analysis

4.00

Undergraduate

Course description not available.

MAT424

Complex Analysis

4.00

Undergraduate

A Major Elective for B.Sc. (Research) Mathematics.
Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)
Prerequisites: MAT 221 Real Analysis II
Overview: This course covers the basic principles of differentiation and integration with complex numbers. Topics will be taught in a computational and geometric way. Knowledge of topology of euclidean space and calculus of several real variables will be assumed.
Detailed Syllabus: Algebraic properties of complex numbers, modulus, complex conjugate, roots of complex numbers, regions. Functions of a complex variable, limits, continuity. Differentiation, Cauchy-Riemann equations, harmonic functions, polar coordinates. Exponential function, logarithm, branch and derivative of logarithm, complex exponents, trigonometric functions, hyperbolic functions, inverse hyperbolic functions. Derivatives of curve w(t) in complex plane, Definite integral of functions w(t), Contours, Contour Integrals, Antiderivatives, Modulus of Contour integrals, Cauchy Goursat theorem. Simply and multiply connected domain, Cauchy Integral Formula and applications, Liouville's theorem, maximum modulus principle. Convergence of series, Power Series, Laurent series, Residues, Cauchy's Residue theorem, Singularities, Zeroes of analytic functions, Behaviour of function near singularities.
References: James W Brown and Ruel V Churchill, Complex Variables and Applications, 8th edition, Tata McGraw-Hill, 2009. H A Priestley, Introduction to Complex Analysis, 2nd edition, Oxford University Press. 2003. J Bak and D J Newman, Complex Analysis, 2nd edition, Springer, 2008. M J Ablowitz and A S Fokas, Complex Variables: Introduction and Applications, 2nd edition, Cambridge University Press India, 2006.

MAT624

Complex Analysis

4.00

Graduate

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: 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.
References: 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.

MAT726

Topics in Complex Analysis

4.00

Graduate

Course description not available.