Physics – High Energy Physics – High Energy Physics - Phenomenology
Scientific paper
2006-03-21
Physics
High Energy Physics
High Energy Physics - Phenomenology
18 pages, 6 figures, Latex
Scientific paper
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.
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