We shall cover chapters I--V of
the text, and sections 1,2 of VI if time permits:
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. |