BSc Computer Science and Mathematics with Industrial Experience

This course unit is a natural successor to the second year course units on Complex Analysis. It introduces multivalued functions, analytic continuation and integral transforms, especially Fourier and Laplace transforms. These powerful and effective tools are used to solve many problems involving differential equations. The course is oriented towards applications rather than the theorem/proof style of development.

Students must have taken (MATH20101 OR MATH20142) AND (MATH20401 OR MATH20411)

This course adopts a methods-style approach to build on the powerful tools of complex analysis introduced in MATH20201/20142, studying 'multi-valued' functions and the strange but powerful idea of analytic continuation. The Gamma function is also introduced and studied before the course turns its attention to Integral Transforms, and then uses Fourier and Laplace Transforms to solve some problems involving PDEs.

On successful completion of this course unit students will be able to: 

• Draw the Branch Cut(s) required by functions involving ln z or z^α (where α is not an integer) and evaluate properties of such functions, including finding poles and their residues, and the integrals of these functions in certain suitable cases.
• Perform contour integration around suitable closed contours (including circular and rectangular contours, D-contours, keyhole contours and dumb-bell contours) in order to evaluate certain real, definite integrals.
• Define the process of Analytic Continuation and apply this to certain suitable functions.
• Define the Gamma Function, and state and use its properties.
• Define Fourier and Laplace Transforms and their inverses, and use their properties to solve certain suitable PDEs.

• Regular Functions: Regular functions of complex z including the multivalued functions lnz and za. Branch lines and branch points. Functions with finite branch lines. [3]
• Contour Integrals: Revision of contour integrals, Cauchy's theorem, Cauchy's integral formula and the residue theorem. Evaluation of residues. Liouville's theorem. [3]
• Real Definite Integrals:. Evaluation of real definite integrals by complex contour methods, especially those involving multivalued functions of z. Deduction of new integrals from known ones by shift of contour. [4]
• Analytic Continuation:. Examples of regular functions defined by series or integrals and their analytic continuations. Uniqueness of analytic continuations and applications. Continuous continuation theorem and Schwarz's principle. [3]
• The Gamma Function: Definition of G(z) as an integral. The functional relation. Analytic continuation of G(z), its poles and residues. The reflection formula.
• Fourier and Laplace Transforms:. Integral transforms in general. Fourier's integral theorem. Functions defined on [0, â'ž), the Fourier cosine and sine transforms and their inverses. The complex Fourier transform and its inverse. Extension to the case in which the transform variable is complex and the inverse transform is a contour integral. The Laplace transform and its relationship to the complex Fourier transform. The Bromwich integral inversion formula. Examples of all of these. [4]
• Applications of Integral Transforms to Partial Differential Equations:. A simple linear ODE solved by Laplace transform. Initial value problem for the one-dimensional heat equation for the infinite bar. Same for the semi-infinite bar with appropriate end conditions. The semi-infinite bar with prescribed end temperature. Boundary value problems for Laplaceâ€'s equation in an infinite strip. Same for Helmholz's equation if time permits. [5]

• Mid-semester coursework: weighting 20%
• End of semester examination: weighting 80%

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
 

All these books, which are further reading, are to be found in Blue 2 515.9 in the John Rylands Library; there are many others which may be worth browsing over. (All have titles which contain the idea of complex analysis or of functions of a complex variable, so the titles are not given here.) Authors:
Phillips, EG.
Carrier, G.
Le Page, W.
Marsden, J.
Mathews, J & Howell, R.
Priestley, H.
Fisher, S.

This course unit detail provides the framework for delivery in 20/21 and may be subject to change due to any additional Covid-19 impact.  

Please see Blackboard / course unit related emails for any further updates