Complex functions-Limit of a function –continuity of functions- uniform continuity-differentiability and analyticity of function–necessary conditions for differentiability –Sufficient conditions for differentiability –Cauchy-Riemann equation in polar coordinates –Complex functions as a function of z and z ̅ - Examples

Hormonic functions- Elementary and conformal mappings: Bilinear transformations (Mobius transformations)-Transformation w=e z , w = sinz, w = cosz. Conformal Mapping- Examples

Cauchy’s Integral Theorem (except Goursat Lemma) – Cauchy’s Integral formula –Derivatives of analytic function – Moreras Theorem – zeros of a function - Cauchy’s inequality – Liouville’sTheorem – Fundamental Theorem of Algebra.

Taylor’s Theorem – Taylor’s Series – Laurent’s Series – Singular points – Types ofsingularities – Identification ofsingularities

Residue – calculus of residues – Residue theorem –Real definite Integrals - type 1, type – 2 ,type -3 Meromorphic function: Principle of argument and Ruche’s Theorem(only) – Problems.