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ISSN 1088-6826(e) ISSN 0002-9939(p)

On the evaluation of generalized Watson integrals

Author(s): G. S. Joyce; I. J. Zucker
Journal: Proc. Amer. Math. Soc. 133 (2005), 71-81.
MSC (2000): Primary 33-xx
Posted: August 24, 2004
MathSciNet review: 2085155
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Abstract: The triple integrals

$\begin{displaymath}W_1(z_1)=\frac{1}{\pi^3}\int_0^\pi\int_0^\pi\int_0^\pi \frac{... ...cos\theta_2+\cos\theta_2\cos\theta_3+\cos\theta_3\cos\theta_1)}\end{displaymath}$

and

$\begin{displaymath}W_2(z_2)=\frac{1}{\pi^3}\int_0^\pi\int_0^\pi\int_0^\pi \frac{... ...a_3}{1-\frac{z_2}{3}(\cos\theta_1+\cos\theta_2+ \cos\theta_3)},\end{displaymath}$

where

and

are complex variables in suitably defined cut planes, were first evaluated by Watson in 1939 for the special cases

and

, 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

and

. It is also shown that

and

are related by the transformation formula

$\begin{displaymath}W_2(z_2)=(1-z_1)^{1/2}W_1(z_1),\end{displaymath}$

where

$\begin{displaymath}z_2^2=-z_1\left(\frac{3+z_1}{1-z_1}\right).\end{displaymath}$

Thus both of Watson's results for $\{W_j(1)\colon j=1,2\}$ are contained within a single formula for

References:

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