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.                                                                             

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 integration of R(Cosϴ,Sinϴ) in the limit (α,2π+α), f(x) in the limit (-∞+∞), Conformal mappings and Mobius transformations.