620-252 Analysis

Students may only gain credit for one of 620-221 and 620-252.

One of [01]620-112, 620-122, 620-142, [99]620-200, 620-211, and one of 620-113, 620-123, 620-143, [98]620-130, [98]620-132.

Note: [98]620-142 is not sufficient to enrol in this subject.

This subject deals with convergence of sequences and series; elementary topology of the real line; the fundamentals of continuity, differentiability of functions of several real variables; analytic functions of a complex variable; complex derivative; power and Laurent series in complex variables; basic topological concepts in the complex plane; and Cauchy's theorem and its applications. Students completing this subject develop the ability to determine the convergence or otherwise of sequences and series; differentiate functions of a complex variable; calculate contour integrals; work with analytic functions in the cut plane; and apply Cauchy's integral formula and the residue theorem. The subject demonstrates the differences between functions of a real and a complex variable; and the role of complex analytic methods in solving important problems in science and engineering.

Sequences and series topics include standard sequences and series, Cauchy convergence, ratio and n^th root tests, absolute and conditional convergence, re-arrangements, and power series. Continuity topics include continuity and differentiability of functions of several real variables. Functions of a complex variable topics include elementary functions of a complex variable, branches; differentiation, analytic functions, and Cauchy-Riemann equations. Integration topics include line and contour integrals, Cauchy's integral theorem; Laurent series; singularities, poles, Liouville's theorem; residue theorem, limiting contours, and evaluation of integrals using contour integration.

Up to 24 pages of written assignments; a 3-hour end-of-semester written examination and class tests totalling not more than 1.5 hours.