Topics in Analysis(C)

TOPICS IN ANALYSIS (C)
Analysis courses from IB will be helpful, but it is intended to introduce and develop concepts of analysis as required.

Discussion of metric spaces; compactness and completeness. Brouwer’s ﬁxed point theorem. Proof(s) in two dimensions. Equivalent formulations, and applications. The degree of a map. The fundamental theorem of algebra, the Argument Principle for continuous functions, and a topological version of Rouch´e’s Theorem. [6]
The Weierstrass Approximation Theorem. Chebychev polynomials and best uniform approximation. Gaussian quadrature converges for all continuous functions. Review of basic properties of analytic functions. Runge’s Theorem on the polynomial approximation of analytic functions. [8]
Liouville’s proof of the existence of transcendentals. The irrationality of e and π. The continued fraction expansion of real numbers; the continued fraction expansion of e. [4]
Review of countability, topological spaces, and the properties of compact Hausdorﬀ spaces. The Baire category theorem for a complete metric space. Applications. [6]

Appropriate books

A.F. Beardon Complex Analysis: the Argument Principle in Analysis and Topology.
John Wiley & Sons, 1979 E.W. Cheney Introduction to Approximation Theory. AMS, 1999
G.H. Hardy and E.M. Wright An Introduction to the Theory of Numbers. Clarendon Press, Oxford, ﬁfth edition, reprinted 1989
T. Sheil-Small Complex Polynomials. Cambridge University. Press, 2002

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