On the Numerical Evaluation of a Class of Oscillatory Integrals in Worldline Variational Calculations

Abstract

Filon-Simpson quadrature rules are derived for integrals of the type \int_a^b dx f(x) sin(xy)/(xy) and \int_a^b dx f(x) 4 sin^2(xy/2)/(xy)^2 which are needed in applications of the worldline variational approach to Quantum Field Theory. These new integration rules reduce to the standard Simpson rule for y = 0 and are exact for y \to \infty when a = 0 and f(0) \ne 0.The subleading term in the asymptotic expansion is also reproduced more and more precisely when the number of integration points is increased. Tests show that the numerical results are indeed stable over a wide range of y-values whereas usual Gauss-Legendre quadrature rules are more precise at low y but fail completely for large values of y. The associated Filon-Simpson weights are given in terms of sine and cosine integrals and have to be evaluated for each value of y. A Fortran program to calculate them in a fast and accurate manner is available. A detailed comparison with the double exponential method of Ooura and Mori is made.