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

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...

Full description

Saved in:
Bibliographic Details
Published inJournal of computational and applied mathematics Vol. 249; pp. 1 - 8
Main Author Ooura, Takuya
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.09.2013
Subjects
Computational complexity
Double exponential quadrature method
Multiple-precision computation
Numerical quadrature
65B99
65D32
68W05
65Y20
Multiple-precision computation
Numerical quadrature
Double exponential quadrature method
Computational complexity
Online AccessGet full text

Cover

More Information
Summary: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.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2013.02.006