An elliptic integral identity
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- by H. S. Wrigge PDF
- Math. Comp. 27 (1973), 839-840 Request permission
Abstract:
The identity \[ K(\tau ) = \frac {1}{{{{(2\pi )}^{1/2}}}}\int _{ - \infty }^\infty {\int _{ - \infty }^\infty {\exp \left [ { - \frac {1}{2}({x^4} - 2(2{\tau ^2} - 1){x^2}{y^2} + {y^4})} \right ]dx\;dy,} } \] 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
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A. Erdélyi et al., Tables of Integral Transforms. Vol. 1, McGraw-Hill, New York, 1954. MR 15, 868.
- I. M. Ryshik and I. S. Gradstein, Summen-, Produkt- und Integral-Tafeln, VEB Deutscher Verlag der Wissenschaften, Berlin; distributed by Plenum Press, Inc., New York, 1963 (German, with English summary). Zweite, berichtigte Auflage. MR 0158100 M. L. Glasser & V. E. Wood, "A closed form evaluation of the elliptic integral," Math. Comp., v. 25, 1971, pp. 535-536.
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Math. Comp. 27 (1973), 839-840
- MSC: Primary 33A25
- DOI: https://doi.org/10.1090/S0025-5718-1973-0324083-4
- MathSciNet review: 0324083