# ACM40690 Applied Complex Variables

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 closed-form solutions in certain asymptotic limits.

Topics will be drawn from the following (non-exhaustive) list:
[Review of complex analysis] Cauchy-Riemann conditions, Cauchy's integral theorem, calculus of residues, harmonic functions, Jensen's formula.
[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; Airy functions.
[Singular perturbation theory] The WKB approximation in the far field and near turning points, applications of WKB theory in Quantum Mechanics and Fluid Mechanics.
[Special functions] Frobenius's theorem, independent solutions, applications involving special functions.

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Curricular information is subject to change

Learning Outcomes:

On completion of the module, students should be able to

1. Carry out calculations using Laplace transforms, solve ODEs via Laplace-transform methods
2. Evaluate certain integrals in asymptotic limits using the saddle-point method
3. Formulate the solution of ODEs as contour integrals and evaluate these integrals in certain limits
4. Solve ODEs in limiting cases using WKB theory, including turning points
5. Solve ODEs via power-series solutions, understand the analytical properties of these solutions

Student Effort Hours:
Student Effort Type Hours
Lectures

24

Specified Learning Activities

36

Autonomous Student Learning

40

Total

100

Approaches to Teaching and Learning:
Lectures, enquiry and problem-based learning.

These activities constitute the basis for the student’s learning, by engaging the student in actual hard work: listening, writing, studying, solving problems and discussing problems with peers and lecturer.
Requirements, Exclusions and Recommendations

Not applicable to this module.

Module Requisites and Incompatibles
Not applicable to this module.

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade

Not yet recorded.

Carry forward of passed components
No

Resit In Terminal Exam
Spring Yes - 2 Hour