Introductory description

This is the third module in the series Analysis 1, 2, 3 that covers rigorous Analysis. It covers convergence of functions and its applications to Integration, Fourier Series and Complex Analysis.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

• Uniform convergence of sequences and series of functions; Weierstrass M-test
• Application to integration: integrals of limits and series, differentiation under the integral sign
• Fourier series: convergence, Parseval, and Gibbs phenomenon (differentiability and rate of decay of coefficients)
• Complex power series and classical functions (exponential, logarithm, sine and cosine, including periodicity)
• Complex integration, contour integrals and Cauchy’s Theorem
• Applications of Cauchy’s formula to evaluate real integrals
• Laurent series, Calculus of residues

Learning outcomes

By the end of the module, students should be able to:

• Learn how to compute contour integrals: Cauchy's integral formulas and applications
• Understand uniform and pointwise convergence of functions together with properties of the limit function
• Learn the continuity, differentiability and integral of the limit of a uniformly convergent sequence of functions
• Develop working knowledge of complex differentiability (Cauchy-Riemann equations) and complex power series
• Develop understanding of Fourier Series including Gibbs phenomenon

Subject specific skills

• Working knowledge of series and sequences, including the development of the notions of convergence and uniform converge for sequences and series of functions.
• Good understanding of Fourier series, including their convergence, Parseval's identity and Gibbs phenomenon.
• Working knowledge of Complex Analysis, including power series, exponential and circular maps, contour integration.
• Mastery of applications of Cauchy's formula to compute integrals in R.

Transferable skills

Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.