Complex Analysis

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• Understand the main applications of the Gamma functions and derive similar statements
• 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
• Compute logarithms and inverse trigonometric functions and calculate Taylor and Laurent series

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

Lectures, tutorial, private study

Summative

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

Referral

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

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.