Resumen de Fast computation of Goursat�s infinite integral with very high accuracy

Takuya Ooura

• We propose an efficient computation method for the infinite integral �Ï ¡Û 0 xdx/(1 + x6 sin2 x), whose integrand contains a series of spikes, approximately £k 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.