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I need to calculate elliptic integrals with singularities, up to a huge number of digits (250-1000). The problem is that Wolfram Mathematica can't do so many digits, and Pari intnum doesn't handle singularities as it seems.

This is an example of what I have to integrate written in latex:

$$\int_4^{+\infty}\frac {dx} {2 \sqrt{x^3 - 112x + 400}}$$

What software and command should I use? Thanks!

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  • $\begingroup$ I don't know what you mean by "Wolfram Mathematica can't do so many digits". I can do 2500 digits in a blink of an eye. (you Integrate[f[x], {x, 4, Infinity}] first, then do N[%, 1000], or whatever) $\endgroup$
    – Igor Rivin
    Commented Sep 21, 2013 at 19:09
  • $\begingroup$ Actually the elliptic curve routines in gp can do this (in addition to the many other useful features like finding the conductor, which here is the famous $5077$). If you set the desired precision with \p and then say E = ellinit([0,0,0,-112,400]) then E[15] and E[16] are the real and imaginary periods, and ellztopoint([4,4]) will give you the desired integral (or maybe twice that integral) up to some period or half-period, etc. $\endgroup$
    – Noam D. Elkies
    Commented Sep 21, 2013 at 20:04

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You can use PARI/GP for high-precision evaluations of elliptic integrals, with boundary singularities, but it requires some preprocessing, as explained here:

Jörg Waldvogel, Computing Integrals of Analytic Functions to High Precision (2008)

We present a set of numerical quadrature algorithms which typically show exponential convergence for analytic integrands, even in the presence of integrable boundary singularities. The algorithms are based on mapping the integration interval onto the entire real axis, together with suitable transformations of the integrand, preferably to a doubly-exponentially decaying function. The transformed integrals are approximated efficiently by the trapezoidal rule; the approximation error may be analyzed by means of Fourier theory. This method results in a practicable algorithm for computing analytic integrals to a precision of hundreds – or thousands – of digits. An almost fully automated implementation in PARI/GP is given.

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  • $\begingroup$ I read the explanation. But I don't get which command should I use in Pari and how? I used just intnum, and the result was not correct. $\endgroup$
    – integral2
    Commented Sep 21, 2013 at 19:07
  • $\begingroup$ thanks for the help. I see the link to the article now. I will read the article, I hope I will get it. $\endgroup$
    – integral2
    Commented Sep 21, 2013 at 19:08

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