Mathematics 3

Module aims

This module introduces the basic concepts of variational calculus and their applications. It also introduces the theory of complex variables and techniques based on complex variables. Applications include derivation of potential flow solutions around complex 2D shapes such as Joukowski aerofoils, and basic 2D design of hypersonic nosecones.

Learning outcomes

On successfully completing this module, you should be able to: 1. Formulate and solve the Euler-Lagrange equations;  2. Apply the basic theory of functions of a complex variable;  3. Derive Cauchy’s theorem and evaluate the integral of a simple complex function around a curve in the complex plane;  4. Derive power series expansions of complex functions about singular points of the functions;  5. Distinguish between the different types of singularities that can arise in the complex plane;  6. Derive Cauchy’s residue theorem and use it to evaluate real integrals over finite and infinite ranges; 7. Calculate inverse Laplace transforms using the residue theorem and apply this technique to solve certain partial differential equations;  8. Use the ideas of conformal mappings to solve equations in geometrically complicated domains (e.g. to determine the inviscid flow field around a Joukowski aerofoil).

Module syllabus

Calculus of variations:  Euler-Lagrange equations; problems with constraints. Functions of a complex variable:  Revision of complex numbers: triangle inequality, polar coordinate representation, curves in the complex plane. Continuity and differentiability of complex functions: analyticity, the Cauchy-Riemann equations. Definitions and properties of elementary complex functions. Branches and branch points. Complex line integrals: definition and properties. Cauchy’s integral theorem and its consequences. Cauchy’s integral formula. Complex power series: Taylor series, Laurent series. Classification of singularities in the complex plane: poles, residues and essential singularities. The residue theorem: contour integration, evaluation of real integrals. Laplace transforms: Revision of basic properties. Derivation of complex inversion formula. Use of contour integration. Application to differential equations. Conformal Mapping: Application to Laplace’s equation. The Joukowski transformation.  


Teaching methods

The module will be delivered primarily through large-class lectures introducing the key concepts and methods, supported by a variety of delivery methods combining the traditional and the technological.  The content is presented via a combination of slides, whiteboard and visualizer.

Learning will be reinforced through tutorial question sheets and coursework.


This module presents opportunities for both formative and summative assessment.  

You will be formatively assessed through tutorial sessions, as well as a marked coursework that is not for credit.

You will have additional opportunities to self-assess your learning via tutorial problem sheets. 

You will be summatively assessed by a written closed-book examination at the end of the module.

Assessment type Assessment description Weighting Pass mark
Examination Closed-book written examination 100% 50%
You will receive feedback following the coursework submission.
You will receive feedback on examinations in the form of an examination feedback report on the performance of the entire cohort.
You will receive feedback on your performance whilst undertaking tutorial exercises, during which you will also receive instruction on the correct solution to tutorial problems.
Further individual feedback will be available to you on request via this module’s online feedback forum, through staff office hours and discussions with tutors.

Reading list

Module leaders

Dr Philip Ramsden