On the evaluation of generalized Watson integrals

Abstract:

The triple integrals \[ W_1(z_1)=\frac {1}{\pi ^3}\int _0^\pi \int _0^\pi \int _0^\pi \frac {\mathrm {d}\theta _1\mathrm {d}\theta _2\mathrm {d}\theta _3}{1-\frac {z_1}{3} (\cos \theta _1\cos \theta _2+\cos \theta _2\cos \theta _3+\cos \theta _3\cos \theta _1)}\] and \[ W_2(z_2)=\frac {1}{\pi ^3}\int _0^\pi \int _0^\pi \int _0^\pi \frac {\mathrm {d}\theta _1\mathrm {d}\theta _2\mathrm {d}\theta _3}{1-\frac {z_2}{3}(\cos \theta _1+\cos \theta _2+ \cos \theta _3)},\] where $z_1$ and $z_2$ are complex variables in suitably defined cut planes, were first evaluated by Watson in 1939 for the special cases $z_1=1$ and $z_2=1$, respectively. In the present paper simple direct methods are used to prove that $\{W_j(z_j)\colon j=1,2\}$ can be expressed in terms of squares of complete elliptic integrals of the first kind for general values of $z_1$ and $z_2$. It is also shown that $W_1(z_1)$ and $W_2(z_2)$ are related by the transformation formula \[ W_2(z_2)=(1-z_1)^{1/2}W_1(z_1),\] where \[ z_2^2=-z_1\left (\frac {3+z_1}{1-z_1}\right ).\] Thus both of Watson’s results for $\{W_j(1)\colon j=1,2\}$ are contained within a single formula for $W_1(z_1)$.