An automatic integration procedure for infinite range integrals involving oscillatory kernels

Abstract

Let the real functionsK(x) andL(x) be such thatM(x)=K(x)+iL(x)=eix g(x), whereg(x) is infinitely differentiable for all largex and is non-oscillatory at infinity. We develop an efficient automatic quadrature procedure for numerically computing the integrals ∫ ∞ K(ωt)f(t) and ∫ ∞ L(ωt)f(t)dt, where the functionf(t) is smooth and nonoscillatory at infinity. One such example for which we also provide numerical results is that for whichK(x)=J ν(x) andL(x)=Y ν(x), whereJ ν(x) andY ν(x) are the Bessel functions of order ν. The procedure involves the use of an automatic scheme for Fourier integrals and the modified W-transformation which is used for computing oscillatory infinite integrals.