Complex Analysis

Unit I: Complex Functions, Limit, continuity, and differentiability, Cauchy-Riemann Equations, Analytic Functions.

Unit II: Power Series, Radius of convergence, Complex Integration, Line or Path Integral, Cauchy Theorem, Cauchy Integral Formula, and their consequences.

Unit III: Elementary Conformal Mappings, Mobius Maps, Maximum Modulus Principle, Liouville Theorem, Fundamental Theorem of Algebra (proof).

Unit IV: Laurent Series, Classification of singularities: Isolated Singularities, Removable, Pole, and Essential, Residue Theorem and evaluation of certain improper integrals.