ACM 40690 Applied Complex Variables (Previously Survey of Applied and Computational Mathematics)

While I am helping to teach this module in 2024, the materials on this page are from a previous academic year, and are posted here for archival purposes only.

Description: This module gives a survey of advanced mathematical methods and their application to problems in physics and more generally, in science and engineering. The aim of the module is to equip students to be well-rounded applied mathematicians, capable of tackling problems using both computational methods and closed-form solutions valid in certain asymptotic limits.

The image is courtesy of Wikipedia. Its creator was Elb2000 and the image was released under a GNU Free Documentation License.

The following is the detailed and finalized list of topics for the academic year 2016-2017:

  • Review of complex analysis: Cauchy-Riemann conditions, Cauchy's integral theorem, calculus of residues, Jensen's formula
  • Harmonic functions: Maximum principle, mean-value property, connection to complex analysis in two dimensions
  • The Green's function for Laplace's equation, arbitrary domains
  • Conformal mapping: Riemann Mapping Theorem, solving Laplace's equation via conformal mapping
  • Laplace transforms: Definition, examples, properties, and inversion via the Bromwich contour
  • Asymptotic methods for integrals: Laplace's method, Watson's lemma, steepest-descent method
  • Writing the solution of an ODE as a contour integral, and the evaluation of the same in asymptotic limits where the steepest-descent method can be used
  • Singular perturbation theory: The WKB approximation in the far field and near turning points
  • Scientific computing with Fortran

Module philosophy: As this is an advanced undergraduate module, heavy emphasis is placed on independent study. There are two lectures per week, and not all of the material in the lecture notes will be covered in class. Your independent study will be guided by reading this material, supplementary material from the recommended textbooks, and by a weekly homework assignments.

Course documents:

Heart-shaped curves

For anyone needing a last-minute (mathematical) Valentine's card, a Latex production file can be downloaded here. For a preview (in pdf) see here.