Course Overview: The theory of functions of a complex variable is a rewarding branch of mathematics to study at the undergraduate level with a good balance between general theory and examples. It occupies a central position in mathematics with links to analysis, algebra, number theory, potential theory, geometry, and topology, and generates a number of powerful techniques (for example, evaluation of integrals) with applications in many aspects of both pure and applied mathematics and other disciplines, particularly the physical sciences.
In these lectures, we begin by introducing students to the theory of (holomorphic) functions of a complex variable.
The central aim of the lectures is to present Cauchy's Theorem and its consequences, particularly series expansions of holomorphic functions, the calculus of residues and its applications.
Learning Outcomes: Students will have become familiar with the main concepts of Complex Analysis. They will have grasped a deeper understanding of differentiation and integration in this setting and will know the tools and results of Complex Analysis including Cauchy's Theorem, Cauchy's integral formula, Liouville's Theorem, Laurent's expansion and the theory of residues.
Course Synopsis: Complex differentiation. Holomorphic functions. Cauchy-Riemann equations (different versions). Real and imaginary parts of a holomorphic function are harmonic.
Recap on power series and differentiation of power series. Exponential function and logarithm function. Fractional powers — examples of multifunctions. The use of cuts as a method of defining a branch of a multifunction.
Path integration. Winding numbers. Cauchy's Theorem (partial proof only). Homology form of Cauchy’s Theorem (sketch of proof only — students referred to various texts for proof.) Fundamental Theorem of Calculus in the path integral/holomorphic situation.
Cauchy's Integral formulae. Taylor expansion. Liouville's Theorem. Morera's Theorem. Identity Theorem.
Laurent's expansion. Classification of isolated singularities. Calculation of principal parts, particularly residues. The argument principle and applications.
Residue Theorem. Evaluation of integrals by the method of residues (main examples only but to include the use of Jordan's Lemma and simple poles on the contour of integration).
