Level of study Third/Final year
Credit value 20
Semester Students may study either 1, 2 or both.
Pre-requisite modules 22497,22507
Semester 1: This module continues the study of complex-valued functions of a complex variable. Analytic functions have more structure that their real counterparts, the differentiable functions, and this extra structure results in a fascinating theory and widespread applications in other areas of mathematics and beyond. The course touches on both these aspects: conformal maps, which have many applications, are discussed, and the techniques of contour integration are extended to deal with functions involving logarithms and roots, while the theory and structure of analytic functions are used to give simple and elegant proofs of the Fundamental Theorem of Algebra.
Semester 2: This module also introduces metric and topological spaces as abstract settings for the study of analytical concepts such as convergence and continuity. This generalization allows one, for example, to regard functions as points of a space and to consider various ways in which the function can be the limit of other functions. Extra structure is introduced: compactness, for instance, is shown to be the proper generalization of the closed bounded intervals often used in analysis on the real line. The course may end with an application: for example, using topological techniques, one can prove the existence of continuous on the real line which fail to be differentiable at any point.