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On the evaluation of generalized Watson integrals


Authors: G. S. Joyce and I. J. Zucker
Journal: Proc. Amer. Math. Soc. 133 (2005), 71-81
MSC (2000): Primary 33-xx
DOI: https://doi.org/10.1090/S0002-9939-04-07447-7
Published electronically: 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 $z_1$ 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

\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 $W_1(z_1)$.


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Additional Information

G. S. Joyce
Affiliation: Wheatstone Physics Laboratory, King’s College, University of London, Strand, London WC2R 2LS, United Kingdom
Email: gsj@maxwell.ph.kcl.ac.uk

I. J. Zucker
Affiliation: Wheatstone Physics Laboratory, King’s College, University of London, Strand, London WC2R 2LS, United Kingdom
Email: jz@maxwell.ph.kcl.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-04-07447-7
Received by editor(s): March 13, 2003
Published electronically: August 24, 2004
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2004 American Mathematical Society

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