On the evaluation of generalized Watson integrals
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- by G. S. Joyce and I. J. Zucker PDF
- Proc. Amer. Math. Soc. 133 (2005), 71-81 Request permission
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 $W_1(z_1)$.References
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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
- Received by editor(s): March 13, 2003
- Published electronically: August 24, 2004
- Communicated by: Jonathan M. Borwein
- © Copyright 2004 American Mathematical Society
- 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
- MathSciNet review: 2085155