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Numerical evaluation of Goursat’s infinite integral

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Abstract

The infinite integral \(\int_0^{\infty}x\,dx/(1+x^6\sin^2x)\) converges but is hard to evaluate because the integrand f(x) = x/(1 + x 6sin2 x) is a non-convergent and unbounded function, indeed f(kπ) = kπ→ ∞ (k→ ∞). We present an efficient method to evaluate the above integral in high accuracy and actually obtain an approximate value in up to 73 significant digits on an octuple precision system in C++.

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Authors and Affiliations

  1. Chukyo University, Toyota, Aichi, Japan

    Yasuyo Hatano

  2. Nagoya University, Nagoya, Japan

    Ichizo Ninomiya

  3. Nanzan University, Seto, Aichi, Japan

    Hiroshi Sugiura

  4. University of Fukui, Fukui, 910-8507, Japan

    Takemitsu Hasegawa

Authors
  1. Yasuyo Hatano
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  2. Ichizo Ninomiya
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  3. Hiroshi Sugiura
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  4. Takemitsu Hasegawa
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Corresponding author

Correspondence to Takemitsu Hasegawa.

Additional information

Dedicated to the memory of Professor Hideo Toda.

Technical details omitted in this paper are given in [12].

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Hatano, Y., Ninomiya, I., Sugiura, H. et al. Numerical evaluation of Goursat’s infinite integral. Numer Algor 52, 213–224 (2009). https://doi.org/10.1007/s11075-009-9265-4

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  • DOI: https://doi.org/10.1007/s11075-009-9265-4

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