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, [05]620-192, [05]620-194, 620-211; and one of 620-113, 620-123, 620-143, [05]620-193.

This subject deals with convergence of sequences and series; elementary topology of the real line; the fundamentals of continuity, and 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, and Cauchy's integral theorem; Laurent series; singularities, poles and Liouville's theorem; and residue theorem, limiting contours, and evaluation of integrals using contour integration.

Up to 36 pages of written assignments due during the semester (0% or 15%); a 3-hour written examination in the examination period (85% or 100%). The relative weighting of the examination and the total assignment mark will be chosen so as to maximise the student's final mark.