11
$\begingroup$

How can we prove $$I:=\int_{\sqrt{33}}^\infty\frac{dx}{\sqrt{x^3-11x^2+11x+121}}\\=\frac1{6\sqrt2\pi^2}\Gamma(1/11)\Gamma(3/11)\Gamma(4/11)\Gamma(5/11)\Gamma(9/11)?$$

Thoughts of this integral
This integral is in the form $$\int\frac{1}{\sqrt{P(x)}}dx,$$where $\deg P=3$. Therefore, this integral is an elliptic integral.
Also, I believe this integral is strongly related to Weierstrass elliptic function $\wp(u)$. In order to find $g_2$ and $g_3$, substitute $x=t+11/3$ to get $$I=2\int_{\sqrt{33}-11/3}^\infty\frac{dt}{\sqrt{4t^3-352/3t+6776/27}}$$ The question boils down to finding $\wp(I;352/3,-6776/27)$ but I seem to be on the wrong track.

$\endgroup$
1
$\begingroup$

Referring to Zacky’s comment it suffices to turn the cubic denominator into a quadratic one, then perform the Landen transform to come to an elliptic integral of the first kind equivalent to K(k11)_ fjaclot

$\endgroup$

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.