Aims & Learning Objectives:
Aims: The aim of this course is to cover the standard introductory material in the theory of functions of a complex variable and to cover complex function theory up to Cauchy's Residue Theorem and its applications.
Students should end up familiar with the theory of functions of a complex variable and be capable of calculating and justifying power series, Laurent series, contour integrals and applying them.
Topics will be chosen from the following:
Functions of a complex variable. Continuity. Complex series and power series. Circle of convergence. The complex plane. Regions, paths, simple and closed paths. Path-connectedness. Analyticity and the Cauchy-Riemann equations. Harmonic functions. Cauchy's theorem. Cauchy's Integral Formulae and its application to power series. Isolated zeros. Differentiability of an analytic function. Liouville's Theorem. Zeros, poles and essential singularities. Laurent expansions. Cauchy's Residue Theorem and contour integration. Applications to real definite integrals.