Sometimes Mathematica expresses results of integration or summation in terms of symbolic derivatives of Hypergeometric2F1 function, and cannot further simplify these derivatives using FunctionExpand or FullSimplify. In some cases I was able to express those derivatives in terms of elementary functions and well-known mathematical constants, but it required some manual work and was on case-by-case basis. Now I have a table of about a hundred of derivatives I already dealt with and a function that can automatically replace them by their values. For example, it contains cases like
Derivative[0, 1, 0, 0][Hypergeometric2F1][-1/2, 3/2, 1/2, 1/Sqrt[2]] ==
(Sqrt[1 + Sqrt[2]] (Sqrt[2] Log[3/2 - Sqrt[2]] + 2 (Sqrt[2] + Log[2 + Sqrt[2]]))
- 4 Sqrt[2] ArcTan[Sqrt[1 + Sqrt[2]]])/2^(3/4)
and
Derivative[2, 0, 0, 0][Hypergeometric2F1][0, -3/4, 1, 1] ==
4 π/3 + 13 π^2/12 - 8 Log[2] - 3 π Log[2] + 9 Log[2]^2 - 8 Catalan
I wonder if anybody else tried to solve this problem and found a more general or automated approach to this? Or if anybody has a more comprehensive table of derivatives and is willing to share it?
I can publish my table if anybody is interested (but I haven't kept all calculations that yielded those results).
Answer
The Package HypExp does exactly that. Here is the link to paper for what I believe was the last extension.
After digging around a bit, the package files should be available here ( Edit freely available link)
Several years ago, there has been some work on the simplification of polylogarithms into a Hopt Algebras, which simplifies the reduction of the Hypergeometic functions in a much faster and simpler way, this would only be useful if you have pages of Hypergeometric functions you need to simplify. I believe there is a package for that too, I used it once, but that was several laptops ago, and don't remember its name if I find it I will post here.
Comments
Post a Comment