Synopsis for Analysis

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.

Synopsis

(1-4) Topology of Euclidean space and its subsets, particularly , , . Open sets, closed sets, subspace topology; continuous functions and their characterisation in terms of pre-images of open or closed sets; connected sets, path-connected sets; compact sets, Heine-Borel Theorem.

(5-7) Complex differentiation. Holomorphic functions. Cauchy-Riemann equations (including version). Real and imaginary parts of a holomorphic function are harmonic.

(8-11) Path integration. Power series and differentiation of power series. Exponential function and logarithm function. Fractional powers - examples of multifunctions. (12-13) Cauchy's Theorem. (Sketch of proof only - students referred to various texts for full proof.) Fundamental Theorem of Calculus in the path integral/holomorphic situation.

(14-16) Cauchy's Integral formulae. Taylor expansion. Liouville's Theorem. Identity Theorem. Morera's Theorem

(17-18) Laurent's expansion. Classification of isolated singularities. Calculation of principal parts, particularly residues.

(19-21) Residue Theorem. Evaluation of integrals by the method of residues (straight forward examples only but to include the use of Jordan's Lemma and simple poles on contour of integration).

(22-23) Conformal mapping, Riemann mapping theorem (no proof): Möbius functions, exponential functions, fractional powers; mapping regions (not Christoffel transformations or Jowkowski's transformation).

(24) Summary and outlook.

1. H. A. Priestley, Introduction to Complex Analysis (second edition, Oxford Science Publications, 2003).
2. T. M. Apostol, Mathematical Analysis (Addison–Wesley, 1974)(Chapter 3 for the topology).
3. Reinhold Remmert, Theory of Complex Functions (Springer, 1989) (Graduate Texts in Mathematics 122).
4. Mark J. Ablowitz, Athanassios S. Focas, Complex Variables, Introduction and Applications(2nd edition, Cambridge Texts in Applied Mathematics, 2003).

Further Reading

1. L. Ahlfors, Complex Analysis (McGraw-Hill, 1979).
2. Theodore Gamelin, Complex Analysis (Springer, 2000).
3. E. C. Titchmarsh, The Theory of Functions (2nd edition, Oxford University Press).
4. I. Stewart and D. Tall, Complex Analysis, (CUP, 1983).
5. J.P Gilman, I Krn and R.E Rogriguez: Complex Analysis, (Springer 2007) (Graduate Texts in Mathematics 122.) This book is included for its extra material showing where the subject can lead.