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, 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 language of topology before using it in the exposition of 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.
The course concludes with an account of the conformal properties of holomorphic functions and applications to mapping regions.
Learning Outcomes: Students will have been introduced to point-set topology and will know the central importance of complex variables in 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: Metric Spaces (10 lectures)
Basic definitions: metric spaces, isometries, continuous functions \(\epsilon-\delta\) definition, homeomorphisms, open sets, closed sets. Examples of metric spaces, including metrics derived from a norm on a real vector space, particularly \(l^1, l^2, l^\infty\) norms on \(\mathbb{R}^n\), the sup norm on the bounded real-valued functions on a set, and on the bounded continuous real-valued functions on a metric space. The characterisation of continuity in terms of the pre-image of open sets or closed sets. The limit of a sequence of points in a metric space. A subset of a metric space inherits a metric. Discussion of open and closed sets in subspaces. The closure of a subset of a metric space. [3]
Completeness (but not completion). Completeness of the space of bounded real-valued functions on a set, equipped with the norm, and the completeness of the space of bounded continuous real-valued functions on a metric space, equipped with the metric. Lipschitz maps and contractions. Contraction Mapping Theorem. [2.5]
Connected metric spaces, path-connectedness. Closure of a connected space is connected, union of connected sets is connected if there is a non-empty intersection, continuous image of a connected space is connected. Path-connectedness implies connectedness. Connected open subset of a normed vector space is path-connected. [2]
Definition of sequential compactness and proof of basic properties of compact sets. Preservation of compactness under continuous maps, equivalence of continuity and uniform continuity for functions on a compact set. Equivalence of sequential compactness with being complete and totally bounded. The Arzela-Ascoli theorem (proof non-examinable). Open cover definition of compactness. Heine-Borel (for the interval only) and proof that compactness implies sequential compactness (statement of the converse only). [2.5]
Complex Analysis (22 lectures)
Basic geometry and topology of the complex plane, including the equations of lines and circles. Extended complex plane, Riemann sphere, stereographic projection. Möbius transformations acting on the extended complex plane. Möbius transformations take circlines to circlines. [3]
Complex differentiation. Holomorphic functions. Cauchy-Riemann equations (including \(z,\bar{z}\) version). Real and imaginary parts of a holomorphic function are harmonic. [2]
Recap on power series and differentiation of power series. Exponential function and logarithm function. Fractional powers — examples of multifunctions. The use of cuts as method of defining a branch of a multifunction. [3]
Path integration. Cauchy's Theorem. (Sketch of proof only — students referred to various texts for proof.) Fundamental Theorem of Calculus in the path integral/holomorphic situation. [2]
Cauchy's Integral formulae. Taylor expansion. Liouville's Theorem. Identity Theorem. Morera's Theorem. [4]
Laurent's expansion. Classification of isolated singularities. Calculation of principal parts, particularly residues. [2]
Residue Theorem. Evaluation of integrals by the method of residues (straightforward examples only but to include the use of Jordan's Lemma and simple poles on contour of integration). [3]
Conformal mappings. Riemann mapping theorem (no proof), Möbius transformations, exponential functions, fractional powers; mapping regions (not Christoffel transformations or Joukowski's transformation). [3]
