Fast computation of Goursat’s infinite integral with very high accuracy

Abstract

We propose an efficient computation method for the infinite integral ∫0∞xdx/(1+x6sin2x), whose integrand contains a series of spikes, approximately π apart, growing taller and narrower as x increases. Computing the value of this integral has been a problem since 1984. We herein demonstrate a method using the Hilbert transform for changing this type of singular function into a smooth function and computing the value of the integral to more than one million significant digits using a superconvergent double exponential quadrature method.