Entry 26

Watson's triple integral $I_1$ is $$\begin{aligned}I_1 &= \frac{1}{\pi^3}\int_0^\pi \int_0^\pi \int_0^\pi \frac{dx\, dy\, dz}{1-\cos x\cos y\cos z}\\ &=\left( \frac{2\sqrt{2}}{\pi}\int_0^{\pi/2}  \frac{dx}{\sqrt{1+\sin^2 x}}\right)^2\\ &= \frac{\Gamma^4(\frac{1}{4})}{4\pi^3}=4\,\eta(i)^4 = 1.393203\dots \end{aligned}$$ where $\eta(\tau)$ is the Dedekind eta function and which was discussed in Entry 25. I found a nice Ramanujan/Chudnovsky-type formula for $I_1$ using the golden ratio $\phi=\tfrac{1+\sqrt{5}}{2}$ $$I_1 =\frac{25\phi^6}{\sqrt{\phi^{24}-4}}\sum_{n=0}^\infty \frac{(6n)!}{(3n)!\,n!^3} \left(\frac{-\phi^{16}}{4(\phi^{24}-4)}\right)^{3n}=1.393203\dots$$ Notice that the $24$th power of the golden ratio is off by $4$.