1
$\begingroup$

I would like to calculate the integral $$\int_0^{2\pi} dx \sin^6\left(\frac{x}{2}\right) F\left(\frac{4-n}{2}, \frac{4+n}{2}, \frac{1}{2}, \cos^2 \frac{x}{2} \right)^2$$ where $F$ is the hypergeometric function (Hypergeometric2F1 in Mathematica). One can see in Mathematica that this integral has some well-defined analytic solution when $n$ is an even integer, excluding $0, -2, 2$ (this is related to the fact that the hypergeometric function is a simple polynomial in these cases). For example, $n=4$ gives $5\pi/8$ and $n=6$ gives $45\pi/32$. I would like to obtain an analytic expression for this integral - however, Mathematica obviously cannot solve this for general $n$. I've tried using Assumptions in the integral, but Mathematica still cannot solve the expression for variable $n$.

$\endgroup$
  • 1
    $\begingroup$ Try replacing $n$ with $2 m$ where $m$ is any integer. $\endgroup$ – flinty Jun 3 at 19:33
  • $\begingroup$ This doesn't work. I don't think the issue is with $n$ being an even integer, but rather the integer assumption more generally - my guess is that Mathematica doesn't try to simplify the hypergeometric function when it assumes $m$ is an integer. $\endgroup$ – Henry Shackleton Jun 3 at 19:46
  • $\begingroup$ Yes, I just tried it, also tweaking with Refine. Disappointing it cannot simplify. $\endgroup$ – flinty Jun 3 at 19:48
8
$\begingroup$
A = Table[{n, 
      Integrate[Sin[x/2]^6 Hypergeometric2F1[(4-n)/2, (4+n)/2, 1/2, Cos[x/2]^2]^2,
                {x, 0, 2π}]}, {n, 4, 20, 2}]

(*    {{4, 5π/8}, {6, 45π/32}, {8, 33π/16}, {10, 429π/160},
       {12, 1287π/392}, {14, 12155π/3136}, {16, 20995π/4704},
       {18, 323π/64}, {20, 7429π/1320}}                          *)

FindSequenceFunction[A, n] // FullSimplify

(*    (2 (9 - 10 n^2 + n^4) π)/(7 n (-4 + n^2))    *)
| improve this answer | |
$\endgroup$
  • $\begingroup$ Wow, I wasn't aware of this function. Thanks! $\endgroup$ – Henry Shackleton Jun 3 at 19:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.