arb
arb copied to clipboard
flintlib
AppellF1 function
The "next higher" level above pFq, the AppellF1 function encompasses incomplete elliptic integrals F and E as well as the complete elliptic integral Π, according to https://en.wikipedia.org/wiki/Appell_series and are needed when expressing solutions to some "simple" integrals like
integrate((a+b*x)^m * (c+d*x)^n * (e+f*x)^p, x)
integrate((a+b*x)^m * (c+d*x^2)^n, x)
integrate((a+b*x)^m * (c+d*x^p)^q, x)
The incomplete elliptic integrals are now implemented separately.
Are there any current plans for the Appell functions (or just F1) to be included in Arb?
Certainly that's an interesting function to include. I don't know how to implement it for |x|, |y| > 1-eps, though.
Hi Fredrik -- I have no particular horse in this race, but if you're truly interested in this, a co-worker of mine has written this for Maple -- but with no bounds on the result, just doing a best-effort attempt at finding a value close to the true value. That might be a starting point. He presented this stuff at ACA 2017, but I don't think there was a paper. I could put you in touch with him, and/or you can read a summary of his work on page 27 and further of this PDF: https://www.mapleprimes.com/DocumentFiles/208439_post/Appell_Functions.pdf.
