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ACM40690

Academic Year 2024/2025

Applied Complex Variables (ACM40690)

Subject:
Applied & Computational Maths
College:
Science
School:
Mathematics & Statistics
Level:
4 (Masters)
Credits:
5
Module Coordinator:
Dr Graham Benham
Trimester:
Autumn
Mode of Delivery:
On Campus
Internship Module:
No
How will I be graded?
Letter grades

Curricular information is subject to change.

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.

About this Module

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
Specified Learning Activities

36
Autonomous Student Learning

40
Lectures

24
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 Component Scale Must Pass Component % of Final Grade In Module Component Repeat Offered
Exam (In-person): Class test Week 1, Week 2, Week 3, Week 4, Week 5, Week 6, Week 7, Week 8, Week 9, Week 10, Week 11, Week 12 Standard conversion grade scale 40% No
20
 No
Assignment(Including Essay): Assignment Week 1, Week 2, Week 3, Week 4, Week 5, Week 6, Week 7, Week 8, Week 9, Week 10, Week 11, Week 12 Standard conversion grade scale 40% No
10
 No
Exam (In-person): Final exam End of trimester
Duration:
2 hr(s)
 Standard conversion grade scale 40% No
70
 No

Carry forward of passed components
No
 

Resit In Terminal Exam
Spring Yes - 2 Hour
Please see Student Jargon Buster for more information about remediation types and timing. 

Feedback Strategy/Strategies

• Group/class feedback, post-assessment

How will my Feedback be Delivered?

This is implemented by the lecturer, who will go through solutions of selected problems.

Timetabling information is displayed only for guidance purposes, relates to the current Academic Year only and is subject to change.
Autumn Lecture Offering 1 Week(s) - Autumn: All Weeks Tues 09:00 - 09:50
Autumn Lecture Offering 1 Week(s) - Autumn: All Weeks Tues 10:00 - 10:50
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