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
Pre-requisites: MATH1048 AND MATH2039
Aims and Objectives
Having successfully completed this module you will be able to:
- Compute logarithms and inverse trigonometric functions and calculate Taylor and Laurent series
- Apply the Mittag-Leffler Theorem and the Weierstrass Product Theorem and understand their proof
- Use complex analysis techniques such as the residue theorem to evaluate real integrals
- Apply (the proof of) Cauchy’s Theorem and Cauchy’s Integral Formula
- Understand the main applications of the Gamma functions and derive similar statements
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
|Preparation for scheduled sessions||12|
|Completion of assessment task||25|
|Wider reading or practice||10|
|Total study time||150|
Resources & Reading list
John B. Conway. Functions of One Complex Variable. Springer-Verlag.
H. A. Priestley. Introduction to Complex Analysis. Oxford Science Publication.
J W Brown, R V Churchill. Complex Variables and Applications. McGraw Hill.
M Beck, G Marchesi, D Pixton, L Sabalka. A First Course in Complex Analysis.
Lars Ahlfors (1979). Complex Analysis. McGraw Hill.
This is how we’ll formally assess what you have learned in this module.
This is how we’ll assess you if you don’t meet the criteria to pass this module.
Repeat type: Internal & External