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 | References | Similar Articles | Additional Information

Abstract: The triple integrals

and

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 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

where

Thus both of Watson's results for are contained within a

*single*formula for .

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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