Gauss's Constant

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

$${1\over M(1,\sqrt{2}\,)} = {2\over \pi}\int_0^1 {1\over\sqrt{1-x^4}}\,dx$$ (1)

$$= {2\over\pi}\int_0^{\pi/2} {d\theta\over\sqrt{1+\sin^2\theta}}$$ (2)

$$= {\sqrt{2}\over\pi} K\left({1\over\sqrt{2}}\right)$$

$$= {1\over(2\pi)^{3/2}} [\Gamma({\textstyle{1\over 4}})]^2$$ (3)

$$= 0.83462684167\ldots,$$ (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

References

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/gauss/gauss.html