The University of Southampton
Courses

# 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).

### 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

TypeHours
Follow-up work25
Lecture36
Preparation for scheduled sessions12
Tutorial12
Revision30
Total study time150

H. A. Priestley. Introduction to 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.

John B. Conway. Functions of One Complex Variable.

Lars Ahlfors (1979). Complex Analysis.

### Assessment

#### Summative

MethodPercentage contribution
Coursework 20%
Examination  (2 hours) 80%

#### Referral

MethodPercentage contribution
Examination  (2 hours) 100%

#### Repeat Information

Repeat type: Internal & External