Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

An elliptic integral identity


Author: H. S. Wrigge
Journal: Math. Comp. 27 (1973), 839-840
MSC: Primary 33A25
DOI: https://doi.org/10.1090/S0025-5718-1973-0324083-4
MathSciNet review: 0324083
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The identity

$\displaystyle K(\tau ) = \frac{1}{{{{(2\pi )}^{1/2}}}}\int_{ - \infty }^\infty ... ... - \frac{1}{2}({x^4} - 2(2{\tau ^2} - 1){x^2}{y^2} + {y^4})} \right]dx\;dy,} } $

where $ K(\tau )$ is the complete elliptic integral of the first kind, is used to prove that $ K(\surd 2 - 1) = {\pi ^{3/2}}{(2 + \surd 2)^{1/2}}/4\Gamma \left( {\frac{5}{8}} \right)\Gamma \left( {\frac{7}{8}} \right)$.

References [Enhancements On Off] (What's this?)

  • [1] A. Erdélyi et al., Tables of Integral Transforms. Vol. 1, McGraw-Hill, New York, 1954. MR 15, 868.
  • [2] I. M. Ryshik and I. S. Gradstein, Summen-, Produkt- und Integral-Tafeln, Zweite, berichtigte Auflage, VEB Deutscher Verlag der Wissenschaften, Berlin; distributed by Plenum Press, Inc., New York, 1963. MR 0158100
  • [3] M. L. Glasser & V. E. Wood, "A closed form evaluation of the elliptic integral," Math. Comp., v. 25, 1971, pp. 535-536.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 33A25

Retrieve articles in all journals with MSC: 33A25


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1973-0324083-4
Article copyright: © Copyright 1973 American Mathematical Society