Complex Analysis | MATH3088 | University of Southampton

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

Linked modules

Pre-requisites: MATH1048 AND MATH2039

Aims and Objectives

Learning Outcomes

Having successfully completed this module you will be able to:

Apply the Mittag-Leffler Theorem and the Weierstrass Product Theorem and understand their proof
Compute logarithms and inverse trigonometric functions and calculate Taylor and Laurent series
Understand the main applications of the Gamma functions and derive similar statements
Apply (the proof of) Cauchy’s Theorem and Cauchy’s Integral Formula
Use complex analysis techniques such as the residue theorem to evaluate real integrals

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

Study time
Type	Hours
Completion of assessment task	25
Follow-up work	25
Wider reading or practice	10
Lecture	36
Preparation for scheduled sessions	12
Revision	30
Tutorial	12
Total study time	150

Resources & Reading list

Textbooks

Lars Ahlfors (1979). Complex Analysis. McGraw Hill.

J W Brown, R V Churchill. Complex Variables and Applications. McGraw Hill.

H. A. Priestley. Introduction to Complex Analysis. Oxford Science Publication.

John B. Conway. Functions of One Complex Variable. Springer-Verlag.

M Beck, G Marchesi, D Pixton, L Sabalka. A First Course in Complex Analysis.

Assessment

Summative

This is how we’ll formally assess what you have learned in this module.

Breakdown
Method	Percentage contribution
Examination	80%
Coursework	20%

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

Breakdown
Method	Percentage contribution
Examination	100%

Repeat

An internal repeat is where you take all of your modules again, including any you passed. An external repeat is where you only re-take the modules you failed.

Breakdown
Method	Percentage contribution
Examination	100%

Repeat Information

Repeat type: Internal & External