Further Complex Variable Theory and Measure and Integration

School of Mathematics

College of Engineering and Physical Sciences

Details

Code 22501

Level of study Third/Final year

Credit value 20

Semester Students may study either 1, 2 or both.

Pre-requisite modules 22497, 22507

Module description

Semester 1: This module continues the study of complex valued functions of a complex variable. Analytic functions have far more structure than their real counterparts, the differentiable functions, and this extra structure results not only in a fascinating theory but also widespread applications in other areas of mathematics and beyond. The course touches on both these aspects: conformal maps, which have many applications in applied mathematics, are discussed and the techniques of contour integration are extended to deal with functions involving logarithms and roots, whilst the theory and structure of analytic functions are investigated and used, for example to give elegant and simple proofs of the Fundamental Theorem of Algebra.

Semester 2: This module also develops the Lebesgue measure, measurable sets and measurable functions. This leads to the definition of the Lebesgue integral and the study of fundamental results including the convergence Theorems.