The complex number system (the complex plane, roots of complex numbers, the extended plane and stereographic projection), Metric Spaces and the topology of C (compactness, continuity and uniform convergence) Elementary properties of analytic functions (power series, analytic functions, Cauchy Riemann Equations, branches of the logarithm, linear fractional transformations, conformal functions), complex integration (Riemann Stieltjes integrals, zeros of analytic functions, Liouville's Theorem, The Fundamental Theorem of Algebra, the index of a closed curve, Cauchy's theorem, the open mapping theorem, Goursat's theorem), Singularities (Laurent expansions, classification of singularities, Casorati-Weierstrass Theorem, residues, the argument principle, Rouche's Theorem). The Maximum Modulus Theorem and Schwarz's lemma.