Complex Analysis

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:

1. Algebraic properties of complex numbers, modulus, complex conjugate, roots of complex numbers, regions.
2. Functions of a complex variable, limits, continuity.
3. Differentiation, Cauchy-Riemann equations, harmonic functions, polar coordinates.
4. Exponential function, logarithm, branch and derivative of logarithm, complex exponents, trigonometric functions, hyperbolic functions, inverse hyperbolic functions.
5. 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.
6. Simply and multiply connected domain, Cauchy Integral Formula and applications, Liouville's theorem, maximum modulus principle.
7. Convergence of series, Power Series, Laurent series, Residues, Cauchy's Residue theorem, Singularities, Zeroes of analytic functions, Behaviour of function near singularities.

References:

1. James W Brown and Ruel V Churchill, Complex Variables and Applications, 8^th edition, Tata McGraw-Hill, 2009.
2. H A Priestley, Introduction to Complex Analysis, 2^nd edition, Oxford University Press. 2003.
3. J Bak and D J Newman, Complex Analysis, 2^nd edition, Springer, 2008.
4. M J Ablowitz and A S Fokas, Complex Variables: Introduction and Applications, 2^nd edition, Cambridge University Press India, 2006.