Complex numbers. Polar coordinates. Functions of a complex variable. Limits. Derivatives of complex functions. Cauchy-Riemann equations. Harmonic functions, harmonic conjugates. Elementary complex functions, Mobius transformations. Contour integrals. Cauchy-Goursat Theorem. Cauchy integral formula. Liouville's theorem. Maximum moduli of functions. Taylor series. Laurent series. Residue theorems. Evaluation of improper integrals. Rouche's theorem.

CILO-1: Identify and apply the Cauchy-Riemann equations to study derivatives. CILO-2: Apply the principle of Mobius transformation to perform various operations. CILO-3: Apply the principle of Liouville’s theorem to analyze functions. CILO-4: Apply the principles of Cauchy-Goursat theorem, Cauchy-Integral formula and Residue theorem to study integrals.