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Math 605, Fall 2001\vskip 10pt
Xavier
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\centerline {\bf SYLLABUS \rm}
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Metric spaces, convergence, continuity, compacteness
and connectedness, uniform convergence. The complex plane and the
Riemann sphere. Power series and their convergence. Analytic functions,
analytic functions as real conformal maps, the real and complex inverse
function theorems, the Cauchy-Riemann equations, Mobius transformations.
Complex integration, various forms of the Cauchy theorem and the Cauchy
integral formula. Cauchy's estimates and applications to Liouville's
theorem and the fundamental theorem of algebra. Theorems
of Morera and Goursat. The open mapping theorem. Isolated singularities.
The Casoratti theorem. The unique continuation principle. Calculation of
integrals by the method of residues. Counting zeros and poles, the
argument principle. The maximum modulus theorem. Rouche's theorem,
Hurwitz's theorem and Schwarz's lemma. Automorphisms of the plane and the
disc. Topology on the space of holomorphic functions, proof of Montel's
theorem. Proof of the Riemann mapping theorem.
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Textbook: Conway's Functions of one complex variable, with eventual use
of Rudin's Real and complex Analysis.
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