Mathematics

Complex Analysis

Module code: G5110
Level 5
15 credits in spring teaching
Teaching method: Lecture, Workshop
Assessment modes: Coursework, Unseen examination

In this module, the topics you will cover will include:

  • holomorphic functions, Cauchy's theorem and its consequences
  • power series, integration, differentiation and analysis of convergence
  • Taylor expansions and circle of convergence
  • Laurent expansions and classification of isolated singularities
  • residue theorem and evaluation of integrals
  • Rouche's theorem and the fundamental theorem of algebra.

Module learning outcomes

  • Substantially increased strength in analysis and analytical rigor, including analytic proofs and constructions from first principles;
  • an algebraic and geometric understanding of complex numbers and complex multiplication including finding explicit roots of simple polynomials;
  • an appreciation for how holomorphic functions refine the notion of real differentiability and the strong consequences of being differentiable in the complex sense;
  • a working knowledge of power series, Laurent series, the residue theorem and the evaluation of real integrals via complex singularities and residues.