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# An Elliptic Integral Identity

H. S. Wrigge
Mathematics of Computation
Vol. 27, No. 124 (Oct., 1973), pp. 839-840
DOI: 10.2307/2005518
Stable URL: http://www.jstor.org/stable/2005518
Page Count: 2
The identity $$K(\tau) = \frac{1}{(2\pi)^{1/2}} \int^\infty_{-\infty} \int^\infty_{-\infty} \exp \lbrack - \frac{1}{2}(x^4 - 2(2\tau^2 - 1)x^2y^2 + y^4)\rbrack dx dy,$$ where $K(\tau)$ is the complete elliptic integral of the first kind, is used to prove that $K(\surd2 - 1) = \pi^{3/2}(2 + \surd2)^{1/2} /4\Gamma(\frac{5}{8})\Gamma(\frac{7}{8})$.