Aims

To introduce students to a variety of special functions using integral representations and linear differential equations as the main technique. To provide students with tools necessary for advanced study in Mathematical Physics

Learning objectives

At the end of the module you should be able to:

• use the general properties of Gamma and Beta functions;
• use methods of studying asymptotic behaviour of functions;
• solve linear differential equations by power series;
• solve linear differential equations by Laplace method;
• use general properties of hypergeometric equation and its solutions;
• use classical orthogonal polynomials

• Solve the Laplace and Poisson equations subject to suitable boundary conditions by a variety of methods.
• Apply these techniques to examples drawn from Mathematical Physics.

Syllabus

Part 1:

• Gamma and Beta functions
• Infinite products
• Asymptotic expansions, method of steepest descent;
• Power series solutions to second-order linear differential equations; singular points and the Frobenius series.
• Hypergeometric equation, properties of its solutions, integral representation for the solutions.
• Confluent hypergeometric equation.
• Hypergeometric polynomials. Classical orthogonal polynomials.

Part 2:

• Potential theory. Laplace and Poisson equations in a domain subject to boundary conditions (Dirichlet, Neumann, Robin). Green functions. Existence and uniqueness of solution. Maximum principle.
• Methods of solution: transform methods, separation of variables, series/special function solutions.
• Helmholtz's theorem and vector potentials. Multipole expansions. Applications in Mathematical Physics (eg. Newtonian gravity, electromagnetism, fluid mechanics).
• Diffusion and wave phenomena in selected examples.

Recommended texts

• G F Simmons, Differential Equations, with Applications and Historical Notes, Tata MacGraw-Hill (paperback) (S7.38 SIM)
• E T Whittaker and G N Watson, A Course of Modern Analysis, Cambridge University Press

• K F Riley, M P Hobson and S J Bence, Mathematical Methods for Physics and Engineering, CUP 2002

Teaching

Spring and Summer Term - 4 lectures per week

Assessment

Three hour unseen examination towards the end of the Summer Term (100%), 

Elective information

This module aims to explore the realm beyond the elementary functions domain. We shall study functions defined by integrals which can not be expressed in terms of elementary functions, such as the celebrated Gamma function. We shall study their properties by means of the complex analysis and differential equations.

Please check prerequisites carefully before asking to take this module as an elective.

Prerequisites

• all Taught in: Differential Equations 
• all Taught in: Complex Analysis
• Vector Calculus