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

• Autores: Takuya Ooura
• Localización: Journal of computational and applied mathematics, ISSN 0377-0427, Vol. 249, Nº 1, 2013, págs. 1-8
• Idioma: inglés
• DOI: 10.1016/j.cam.2013.02.006
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• Resumen

  • 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.