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Gauss's Constant

The Reciprocal of the Arithmetic-Geometric Mean of 1 and $\sqrt{2}$,

$\displaystyle {1\over M(1,\sqrt{2}\,)}$ $\textstyle =$ $\displaystyle {2\over \pi}\int_0^1 {1\over\sqrt{1-x^4}}\,dx$ (1)
  $\textstyle =$ $\displaystyle {2\over\pi}\int_0^{\pi/2} {d\theta\over\sqrt{1+\sin^2\theta}}$ (2)
  $\textstyle =$ $\displaystyle {\sqrt{2}\over\pi} K\left({1\over\sqrt{2}}\right)$  
  $\textstyle =$ $\displaystyle {1\over(2\pi)^{3/2}} [\Gamma({\textstyle{1\over 4}})]^2$ (3)
  $\textstyle =$ $\displaystyle 0.83462684167\ldots,$ (4)

where $K(k)$ is the complete Elliptic Integral of the First Kind and $\Gamma(z)$ is the Gamma Function.

See also Arithmetic-Geometric Mean, Gauss-Kuzmin-Wirsing Constant


Finch, S. ``Favorite Mathematical Constants.''

© 1996-9 Eric W. Weisstein