Math 303, Elements of Analysis II
Usually alternates with Math 208/310

This course continues the study of real analysis from 302. It applies the fundamental concepts you learned there to integration and infinite series, and explores some of the connections between the two topics. It takes a detailed look at Riemann's theory of integration, which you saw on a less theoretical level in calculus, and at its limitations. For example, which functions can be integrated using this theory and which cannot? How do you integrate a function that is defined as a limit, such as the sum of an infinite series? What can be done about the functions that cannot be integrated using this theory? We'll study Lebesgue's theory of integration as a way of improving Riemann's theory.
Along the way we'll consider functions as forming both metric spaces and infinite-dimensional inner product spaces, tying analysis and linear algebra together. This approach is characteristic of the field of functional analysis.

Depending on the time available, we may study how Fourier series can be used to express functions in terms of series of exponentials (or sines and cosines). We may also consider how integration can be used to unify ideas from seemingly different branches of probability, and how Fourier series can describe the sounds produced by musical instruments.

Prerequisite: 302.
Distribution: Mathematical Modeling. Majors can fulfill the major presentation requirement in this course in 2005-06.

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