1.  Introduction:  This second semester course at level 3 follows on from the techniques stream courses of level 2 Advanced Calculus I and II. It is designed to be equally suitable for students taking pure or applied units in their final year.

2.  Timetable Hours, Credits, Assessments:  The unit is of 20 UCU and is taught in the Spring Semester by means of 33 hours of lectures, three a week. No formal support teaching is timetabled, but office hours are advertised and students are encouraged to seek help from the lecturer when required. The Assessment is by coursework set three times through the semester (20%) and a three-hour examination (80%) near the end of the semester.

3.  Overview:  This unit continues the methods stream of the degree programme, introducing techniques relevant to applied mathematics and theoretical physics, but stressing the rigorous mathematical foundations of the subject. The broadly unifying theme of the material covered is the theory of integral transforms such as Laplace and Fourier transforms.

Initially, topics from analysis units studied in levels 1 and 2 will be revised and extended. However, the level 2 Analysis unit is not a pre-requisite for this unit. Convergence of sequences of functions will be considered. A natural question to ask then, is whether a given function can be approximated by a sum of other functions?

The Sturm-Liouville eigenvalue problem leads to the formulation of sequences of mutually orthogonal functions. Sturm and Liouville made some progress in demonstrating the completeness of such sequences, and ad hoc proofs for particular cases can be constructed. However, a rigorous proof for the general result was not given until the beginning of the 20th century, and indeed the search for such a proof constituted one of those problems which give direction to mathematics.

Analysis of eigenfunction expansions leads naturally to Fourier series and hence to the concept of Fourier transforms. Indeed, the inversion of Fourier integrals can be investigated by analogy with convergence of Fourier series. Applications of Fourier and Laplace transforms include the solution of differential equations. A wide range of other applications of Fourier theory will also be considered, ranging from topics in pure mathematics to more physical applications.

Techniques are available for the evaluation of integrals involving large or small parameters, a topic known as asymptotic analysis. Such techniques are particularly suitable for the inversion integrals arising from transform theory.

4.  Recommended Reading:   No one book covers all the course. References are provided at the end of each section. Books which prove useful to several parts of the course include Arfken, Korner and Hinch.

Arfken 'Mathematical Methods for Physicists'

Lighthill 'An introduction to Fourier analysis & generalised functions'

Korner 'Fourier Analysis'

Kreyszig 'Advanced Engineering Mathematics'

Hinch 'Perturbation Methods'

Standard results on convergence of series & integrals including Weierstrass M-test. Convergence of functions: pointwise and uniform convergence. (5 lectures)

Introduction and history. Discussion and proof of results of Fourier, Dirichlet & Cauchy. Examples of Fourier Series illustrating these results - eg Gibbs Phenomena. Further discussion of convergence of Fourier Series, reference to work of Du Bois-Reymond as possible precursor to more detailed study later in the unit. (6 lectures)

Adjoint operators. Hermititian operators and orthogonality of eigenfunctions. Orthogonal polynomials. Examples including Legendre & Chebyshev polynomials. (4 lectures)

Including inversion theorems and convolution. Relevance of theory developed for Fourier series. Examples of applications. Possible discussion of use of Fourier transforms in numerics (FFTs). (5 lectures)

Emphasis on inversions by contour deformation. Examples of application (including integral equations). Inversions involving branch cuts. (5 lectures)

Emphasis on asymptotic evaluation of inverse transform integrals. Contour deformation and method of steepest descent. (5 lectures)

Topics illustrating theoretical underpinning of techniques. More on convergence of Fourier Series. Fejer's Theorem. (3 lectures)