Complex Analysis (Optional Paper)

Functions of a complex Variable, Differentiability and analyticity, Cauchy Riemann Equations, Harmonic functions, Existence of Harmonic conjugate, transcendental functions such as exponential, trigonometric and hyperbolic functions.                                                                            

Power series as an analytic function. Properties of line integrals, Goursat Theorem, Cauchy theorem, Consequence of simply connectivity, Index of a closed curve, Cauchy’s integral formula.

Morera’s theorem, Liouville’s theorem, Fundamental theorem of Algebra, Taylor’s theorem, Laurent series, singularities, Classification of singularities, Zeros of Analytic functions

Maximum modulus theorem, Minimum modulus theorem, Hadamard three circle theorem, Schwarz’s Lemma, Rouche’s theorem, Calculation of residues, Residue theorem

Evaluation of integrals of the form , , Conformal mappings and Mobius transformations.