This thesis is concerned with the evaluation of rapidly oscillatory integrals, that
is integrals in which the integrand has numerous local maxima and minima over
the range of integration. Three numerical integration rules are presented. The
first is suitable for computing rapidly oscillatory integrals with trigonometric oscillations
of the form f(x) exp(irq(x)). The method is demonstrated, empirically,
to be convergent and numerically stable as the order of the formula is increased.
For other forms of oscillatory behaviour, a second approach based on Lagrange's
identity is presented. The technique is suitable for any oscillatory weight function,
provided that it satisfies an ordinary linear differential equation of order
m :2:: 1. The method is shown to encompass Bessel oscillations, trigonometric
oscillations and Fresnel oscillations, and products of these terms. Examples are
included which illustrate the efficiency of the method in practical applications.
Finally, integrals where the integrand is singular and oscillatory are considered.
An extended Clenshaw-Curtis formula is developed for Fourier integrals which
exhibit algebraic and logarithmic singularities. An efficient algorithm is presented
for the practical implementation of the method.
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.