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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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
Keywords
- Infinite integral
- Unbounded integrand
- Contour integral
- Residues
- Numerical evaluation
- High accuracy
- Octuple precision