COMPLEX ANALYSIS- TOPICS FOR UGC NET MATHEMATICAL SCIENCE

1.  Algebra of complex numbers, the complex plane, polynomials, power series, transcendental functions such as exponential, trigonometric and hyperbolic functions.
2.  Analytic functions, Cauchy-Riemann equations.
3. Contour integral, Cauchy’s theorem, Cauchy’s integral formula, Liouville’s theorem, Maximum modulus principle, Schwarz lemma, Open mapping theorem. 
4. Taylor series, Laurent series, calculus of residues. 
5. Conformal mappings, Mobius transformations.