620-252 Analysis

Note

Students may only gain credit for one of 620-221, 620-252, (1997 Handbook 618-201, 618-252; 1996 Handbook 618-202).

Credit Points

Coordinator

Prerequisites

One of 620-112, 620-122, 620-200, 620-211 (1997 Handbook 618-112, 618-122, 618-200, 618-211).

Semester

Contact

Subject Description

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

Assessment

Up to 24 pages of written assignments and a three-hour end-of-semester written examination.