Analytic functions: Complex functions-Limit of function –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.

Harmonic functions- Examples. Elementary and conformal mappings: Bilinear transformations (Mobius transformations)-Transformation w=e z , w = sin z, w = cos z. - Conformal Mapping- Examples.

Complex integration- 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’s Theorem – Fundamental Theorem of Algebra.

Taylor’s Theorem – Taylor’s Series – Laurent’s Series – Singular points – Types of singularities – Identification of singularities.

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