We shall cover chapters IV of
the text, and sections 1,2 of VI if time permits:
History, 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, CasoratiWeierstrass Theorem, residues, the argument principle,
Rouche's Theorem). The Maximum Modulus Theorem and Schwarz's lemma.
