Equality of two elliptic integrals

Question

by wolfram 1 and wolfram 2

It's true that $\displaystyle \int_0^{\frac{\pi}{2}}\int_0^{\frac{\pi}{2}} \frac{2}{\sqrt{1-\sin^2\theta\sin^2\phi}} d\phi d\theta =\int_0^{\frac{\pi}{2}}\int_0^{\frac{\pi}{2}} \frac{1}{\sqrt{\sin\theta\sin\phi}} d\phi d\theta$ ?

If yes, I need a way to prove the equality of the following two integrals

I tried everything but I am unable to convert into a standard form so How do I solve this problem.

Addition 1: For the second integral $\int_0^{\frac{\pi}{2}}\int_0^{\frac{\pi}{2}} \frac{1}{\sqrt{\sin\theta\sin\phi}} d\phi d\theta =\bigg(\int_0^{\frac{\pi}{2}}\frac{1}{\sqrt{\sin\theta}}d\theta\bigg)^2 $ and we can use that $2 \int^{\frac{\pi}{2}}_{0}\frac{1}{\sqrt{\sin x}} \mathrm dx = \frac{\Gamma(1/2)\Gamma(1/4)}{\Gamma(3/4)} = \frac{\Gamma \left( \frac{1}{4}\right)^2}{\sqrt{2\pi}}$

Addition 2: For the first integral

Let $\displaystyle K(k)=\int_0^{\frac{\pi}{2}}\frac{1}{\sqrt{1-k^2\sin^2 t}}dt$ ( Complete Elliptic Integral of the First Kind). we know that $ \displaystyle K(k)=\frac{\pi}{2}\sum_{n=0}^\infty \left(\frac{(2n)!}{2^{2n}(n!)^2}\right)^2k^{2n}$. Then it's not difficult de show that $\int_0^{\frac{\pi}{2}}\int_0^{\frac{\pi}{2}} \frac{1}{\sqrt{1-\sin^2\theta\sin^2\phi}} d\phi d\theta=(\pi /2 )^2 \sum _{n=0}^{\infty }(\frac{(2n)!}{4^n(n!)^2})^3$.

The egality of the two integrals hold if we can calculte $$\sum _{n=0}^{\infty }(\frac{(2n)!}{4^n(n!)^2})^3$$ Wolfram gives

Addition 3:

This link gives

Answer 1

The equivalence of the elliptic integrals can be shown in the following way: $$\begin{align} \int\limits_0^{\frac{\pi}{2}}\int\limits_0^{\frac{\pi}{2}} \frac{d\phi\, d\theta}{\sqrt{1-\sin^2\theta\sin^2\phi}} & =\int\limits_0^{\frac{\pi}{2}}K(\sin\theta)d\theta\tag1\\ &=\int\limits_0^{1}\frac{K(r)dr}{\sqrt{1-r^2}}\tag2\\ &=\int\limits_0^{1}\frac{K\left(\frac{2\sqrt k}{1+k}\right)dk}{\sqrt{k}(1+k)}\tag3\\ &=\int\limits_0^{1}\frac{K\left(k\right)dk}{\sqrt{k}}\tag4\\ &=\int\limits_0^{1}\frac{dk}{\sqrt{k}} \int\limits_0^{\frac{\pi}{2}} \frac{d\phi}{\sqrt{1-k^2\sin^2\phi}}\tag5\\ &=\int\limits_0^{\frac{\pi}{2}}d\phi\int\limits_0^{1} \frac{dk}{\sqrt{k}\sqrt{1-k^2\sin^2\phi}}\tag6\\ &=\int\limits_0^{\frac{\pi}{2}}d\phi\int\limits_0^{\phi} \frac{d\theta}{\sqrt{\sin\theta\sin\phi}}\tag7\\ &=\frac12\int\limits_0^{\frac{\pi}{2}}\int\limits_0^{\frac{\pi}{2}} \frac{d\phi\, d\theta}{\sqrt{\sin\theta\sin\phi}}.\tag8\\ \end{align}$$

Explanation:

$(1)$: Definition of the complete elliptic integral of the first kind $K(k)=\int\limits_0^{\frac{\pi}{2}}\frac{d\phi}{\sqrt{1-k^2\sin^2\phi}}$.

$(2)$: $\sin\theta\mapsto r$.

$(3)$: $r\mapsto\frac{2\sqrt k}{1+k}$.

$(4)$: Landen's transformation $K(k)=\frac1{1+k}K\left(\frac{2\sqrt k}{1+k}\right)$.

$(5)$: Definition of the complete elliptic integral of the first kind.

$(6)$: Interchange of integration order.

$(7)$: $k\mapsto\frac{\sin\theta}{\sin\phi}$.

$(8)$: Use of the integrand symmetry.

Answer 2

I wish i have found out the following solution (thanks go to Etanche and Jandri).

\begin{align}J&=\int_0^{\frac{\pi}{2}}\int_0^{\frac{\pi}{2}} \frac{1}{\sqrt{1-\sin^2(\theta)\sin^2 \varphi}}d\varphi d\theta\\ &\overset{z\left(\varphi\right)=\arcsin\left(\sin(\theta)\sin \varphi\right)}=\int_0^{\frac{\pi}{2}} \left(\int_0^ \theta\frac{1}{\sqrt{\sin(\theta-z)\sin(\theta+ z)}}dz\right)d\theta\tag1\\ &=\frac{1}{2}\int_0^{\frac{\pi}{2}} \left(\int_{u}^{\pi-u}\frac{1}{\sqrt{\sin u\sin v}}dv\right)du \tag2\\ &=\frac{1}{2}\int_0^{\frac{\pi}{2}} \left(\int_{u}^{\frac{\pi}{2}}\frac{1}{\sqrt{\sin u\sin v}}dv\right)du+\underbrace{\frac{1}{2}\int_0^{\frac{\pi}{2}} \left(\int_{\frac{\pi}{2}}^{\pi-u}\frac{1}{\sqrt{\sin u\sin v}}dv\right)du}_{w=\pi-v}\\ &=\int_0^{\frac{\pi}{2}} \left(\int_{u}^{\frac{\pi}{2}}\frac{1}{\sqrt{\sin u\sin v}}dv\right)du\\ &=\int_0^{\frac{\pi}{2}} \int_0^{\frac{\pi}{2}}\frac{1}{\sqrt{\sin u\sin v}}dudv-\int_0^{\frac{\pi}{2}} \left(\int_{0}^{u}\frac{1}{\sqrt{\sin u\sin v}}dv\right)du\\ &=\int_0^{\frac{\pi}{2}} \int_0^{\frac{\pi}{2}}\frac{1}{\sqrt{\sin u\sin v}}dudv-\int_0^{\frac{\pi}{2}} \frac{1}{\sqrt{\sin u}}\left(\int_0^{u}\frac{1}{\sqrt{\sin v}}dv\right)du\\ &=\int_0^{\frac{\pi}{2}} \int_0^{\frac{\pi}{2}}\frac{1}{\sqrt{\sin u\sin v}}dudv-\frac{1}{2}\left(\int_0^{\frac{\pi}{2}}\frac{1}{\sqrt{\sin v}}dv\right)^2\tag3\\ &=\boxed{\frac{1}{2}\int_0^{\frac{\pi}{2}} \int_0^{\frac{\pi}{2}}\frac{1}{\sqrt{\sin u\sin v}}dudv} \end{align}

$(1)$: $\displaystyle dz=\dfrac{\sqrt{\sin^2\theta-\sin^2 z}}{\sqrt{1-\sin^2 z}}d\varphi$, $\sin^2 a-\sin^2 b=\sin(a-b)\sin(a+b)$

$(2)$: Perform the change of variable $u=\theta-z,v=\theta+z$

$(3)$: $\displaystyle\int_0^{\frac{\pi}{2}} f(x)f^\prime(x)dx=\frac{1}{2}\left(f^2\left(\frac{\pi}{2}\right)-f^2(0)\right)$ and $\displaystyle f(x)=\int_0^{x}\frac{1}{\sqrt{\sin u}}du$

(Edit: problem fixed again)