MATH3088 Complex Analysis
Module Overview
Complex Analysis is the theory of functions in a complex variable. While the initial theory is very similar to Analysis (i.e, the theory of functions in one real variable as seen in the second year), the main theorems provide a surprisingly elegant, foundational and important insight into Analysis and have far-reaching and enlightening applications to functions in real variables. After introducing differentiability of complex functions and contour integrals, the highlights of this module are presented: Cauchy’s Theorem, Cauchy’s Integral Formula and the Residue Theorem. Finally, the theory is applied to prove the Fundamental Theorem of Algebra, to evaluate real integrals, to provide partial-fraction and infinite-product expansions of classical functions and to introduce and study the Gamma function (a fundamental function that for example affords an asymptotic formula for factorials and a computation of the integral of the Gaussian bell curve). One of the pre-requisites for MATH6094
Aims and Objectives
Module Aims
This module will introduce fundamental concepts and theorems in complex analysis with a view towards applications in real analysis and to the gamma function.
Learning Outcomes
Learning Outcomes
Having successfully completed this module you will be able to:
- Apply (the proof of) Cauchy’s Theorem and Cauchy’s Integral Formula
- Compute logarithms and inverse trigonometric functions and calculate Taylor and Laurent series
- Use complex analysis techniques such as the residue theorem to evaluate real integrals
- Apply the Mittag-Leffler Theorem and the Weierstrass Product Theorem and understand their proof
- Understand the main applications of the Gamma functions and derive similar statements
Syllabus
Review: complex numbers; open, closed, compact and connected subsets; continuity Complex differentiable functions and Cauchy-Riemann equations Review of uniform convergence and power series; exponential and trigonometric functions Contours, contour integrals and elementary properties Goursat's theorem and Cauchy’s theorem for convex open subsets Cauchy integral formula (for disks only) Cauchy inequalities, maximum and minimum principle Liouville’s theorem and fundamental theorem of algebra Logarithm, power and inverse trigonometric functions Taylor and Laurent series (only sketch of existence proofs) Singularities, winding numbers of closed contours, residues and the residue theorem Evaluation of real integrals Partial fraction decompositions, Mittag-Leffler theorem Infinite products, Weierstrass product theorem, examples Gamma function (as time permits): Weierstrass' infinite product definition, Euler's reflection formula, Euler's infinite product formula, Euler's integral of the second kind, Legendre's duplication formula, Stirling's formula
Learning and Teaching
Teaching and learning methods
Lectures, tutorial, private study
| Type | Hours |
|---|---|
| Follow-up work | 25 |
| Wider reading or practice | 10 |
| Completion of assessment task | 25 |
| Tutorial | 12 |
| Preparation for scheduled sessions | 12 |
| Lecture | 36 |
| Revision | 30 |
| Total study time | 150 |
Resources & Reading list
Lars Ahlfors (1979). Complex Analysis.
M Beck, G Marchesi, D Pixton, L Sabalka. A First Course in Complex Analysis.
J W Brown, R V Churchill. Complex Variables and Applications.
H. A. Priestley. Introduction to Complex Analysis.
John B. Conway. Functions of One Complex Variable.
Assessment
Summative
| Method | Percentage contribution |
|---|---|
| Coursework | 20% |
| Examination (2 hours) | 80% |
Referral
| Method | Percentage contribution |
|---|---|
| Examination (2 hours) | 100% |
Repeat Information
Repeat type: Internal & External
Linked modules
Pre-requisites: MATH1048 AND MATH2039