Complex Analysis

Polar representation and roots of complex numbers; Spherical representation of extended complex plane; Elementary properties and examples of analytic functions: The exponential, Trigonometric functions, Mobius transformations, Cross ratio; Complex integration: Power series representation of analytic functions, Zeros of analytic functions, Cauchy theorem and integral formula, The index of a point with respect to a closed curve, the general form of Cauchy’s theorem; Open Mapping Theorem; Classification of singularities: Residue theorem and applications; The Argument Principle; The Maximum modulus Principle; Schwarz’s lemma; Phragmen-Lindelof theorem.