Aims & Learning Objectives:
The aim of this course is to develop the theory of functions of a complex variable and to cover complex function theory up to Cauchy's Residue Theorem and its applications.
On completion of the course, students should have mastered the essentials of the theory of functions of a complex variable. They should be capable of justifying, and have mastered the calculation of, power series, Laurent series, contour integrals and, through assessed coursework, their application. They should be able to demonstrate an in-depth understanding of the subject.
Content:
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 Formula 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.
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