For delivery to students completing Stage 1 before September 2016:
Pre-requisite: MA321 (Calculus and Mathematical Modelling), MA322 (Proofs and Numbers), MA323 (Matrices and Probability), MA552 (Analysis), MA588 (Mathematical Techniques and Differential Equations).
For delivery to students completing Stage 1 after September 2016:
Pre-requisite: MAST4004 (Linear Algebra); MAST4010 (Real Analysis 1); MAST5013 (Real Analysis 2); MAST5012 (Ordinary differential equations).
OverviewThis module provides an introduction to the study of orthogonal polynomials and special functions. They are essentially useful mathematical functions with remarkable properties and applications in mathematical physics and other branches of mathematics. Closely related to many branches of analysis, orthogonal polynomials and special functions are related to important problems in approximation theory of functions, the theory of differential, difference and integral equations, whilst having important applications to recent problems in quantum mechanics, mathematical statistics, combinatorics and number theory. The emphasis will be on developing an understanding of the structural, analytical and geometrical properties of orthogonal polynomials and special functions. The module will utilise physical, combinatorial and number theory problems to illustrate the theory and give an insight into a plank of applications, whilst including some recent developments in this field. The development will bring aspects of mathematics as well as computation through the use of MAPLE. The topics covered will include: The hypergeometric functions, the parabolic cylinder functions, the confluent hypergeometric functions (Kummer and Whittaker) explored from their series expansions, analytical and geometrical properties, functional and differential equations; sequences of orthogonal polynomials and their weight functions; study of the classical polynomials and their applications as well as other hypergeometric type polynomials.
This module appears in:
Method of assessment
80% Examination, 20% Coursework
The module does not follow a specific text. However, the following texts cover the material.
R. Askey, Orthogonal Polynomials and Special Functions, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1975
R. Beals and R. Wong, Special Functions – A Graduate Text, Cambridge University Press, Cambridge, 2010
T.S. Chihara, An Introduction to Orthogonal Polynomials, Dover Publ., Mineola, N.Y., 2011
M. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, Cambridge, 2005
F.W.J. Olver, D.W. Lozier, C.W. Clark,R.F. Boisvert, Digital Library of Mathematical Functions, National Institute of Standards and Technology, Gaithersburg, U.S.A., 2010 (http://dlmf.nist.gov)
I.N. Sneddon, Special Functions of Mathematical Physics and Chemistry, 3rd Edition, Longman, London, 1980
G. Szego, Orthogonal Polynomials, 4th Ed., American Mathematical Society, Providence, RI, 1975
The intended subject specific learning outcomes
On successful completion of this module, students will be able to:
a) understand the basic concepts of orthogonal polynomials and special functions;
b) have sound knowledge of inner products in L2-spaces as well as the skills to apply this knowledge to problems in differential and difference equations;
c) understand how to apply the theory of analytical functions, differential and difference equations and asymptotic methods.
The intended generic learning outcomes
On successful completion of the module, the students will have:
a) an enhanced ability to reason and deduce confidently from given definitions and constructions;
b) enhanced knowledge of special functions and their geometric, analytical and asymptotic properties;
c) matured in their problem formulating and solving skills;
d) consolidated their grasp of a wide variety of mathematical skills and methods.