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I need to compute elliptic integrals of the third kind with complex arguments, preferably in C++. Is there code out there to do this? I have discovered the Arb library, but that does much more than I would like to do, and would be much slower for my particular application.

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    $\begingroup$ DLMF would be the obvious place to look, but the site is currently down due to partial US government shutdown. Cephes doesn't offer the elliptic integral of the 3rd kind, GSL and Boost have it but not for complex types. So the well-respected Arb library seems like a good way to go; see also Fredrik Johansson, "Numerical Evaluation of Elliptic Functions, Elliptic Integrals and Modular Forms", ArXiv manuscript 2018. From a brief perusal of relevant literature, a homebrew implementation seems like a daunting task. $\endgroup$ – njuffa Jan 15 at 1:02
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    $\begingroup$ @njuffa this should be an answer, not a comment ) $\endgroup$ – Anton Menshov Jan 15 at 3:32
  • $\begingroup$ @AntonMenshov IMHO it would be a weak answer, at best. The question asks for the best way to compute elliptic integrals, and I am afraid I have zero insight into that; these are not functions I have previously implemented or researched. I have not used the Arb library either, but have been aware of it for a number of years by reputation. $\endgroup$ – njuffa Jan 15 at 3:39
  • $\begingroup$ Have you done performance testing on Arb? That guy is super smart. I wouldn't put it out of the question that Arb does more than you want, and is fast enough for your application. $\endgroup$ – user14717 Jan 15 at 17:27
  • $\begingroup$ @Danny If I read the software index at DLMF correctly, none of the libraries listed provides elliptic integrals of the third kind with complex arguments. Curiously, the index doesn't include the Arb library, although DLMF is being updated continuously, with the last update occurring in December 2018 just before the government shutdown. $\endgroup$ – njuffa Jan 28 at 20:50

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