Gauss's constant

In mathematics, Gauss's constant, denoted by G, is defined as the reciprocal of the arithmetic–geometric mean of 1 and the square root of 2:

G={\frac {1}{\operatorname {agm} \left(1,{\sqrt {2}}\right)}}=0.8346268\dots .

The constant is named after Carl Friedrich Gauss, who on 1799[1] discovered that

G={\frac {2}{\pi }}\int _{0}^{1}{\frac {dx}{\sqrt {1-x^{4}}}}

so that

G={\frac {1}{2\pi }}\mathrm {B} \left({\tfrac {1}{4}},{\tfrac {1}{2}}\right)

where Β denotes the beta function.

Relations to other constants

Gauss's constant may be used to express the gamma function at argument 1/4:

\Gamma \left({\tfrac {1}{4}}\right)={\sqrt {2G{\sqrt {2\pi ^{3}}}}}

Alternatively,

G={\frac {\left[\Gamma \left({\tfrac {1}{4}}\right)\right]^{2}}{2{\sqrt {2\pi ^{3}}}}}

and since $π$ and Γ(1/4) are algebraically independent, Gauss's constant is transcendental.

Gauss's constant may be used in the definition of the lemniscate constants, the first of which is:

L_{1}=\pi G

and the second constant:

L_{2}={\frac {1}{2G}}

which arise in finding the arc length of a lemniscate. Both constants were proven to be transcendental.[2]

A formula for G in terms of Jacobi theta functions is given by

G=\vartheta _{01}^{2}\left(e^{-\pi }\right)

as well as the rapidly converging series

G={\sqrt[{4}]{32}}e^{-{\frac {\pi }{3}}}\left(\sum _{n=-\infty }^{\infty }(-1)^{n}e^{-2n\pi (3n+1)}\right)^{2}.

The constant is also given by the infinite product

G=\prod _{m=1}^{\infty }\tanh ^{2}\left({\frac {\pi m}{2}}\right).

It appears in the evaluation of the integrals

{\frac {1}{G}}=\int _{0}^{\frac {\pi }{2}}{\sqrt {\sin(x)}}\,dx=\int _{0}^{\frac {\pi }{2}}{\sqrt {\cos(x)}}\,dx

G=\int _{0}^{\infty }{\frac {dx}{\sqrt {\cosh(\pi x)}}}

Gauss' constant as a continued fraction is [0, 1, 5, 21, 3, 4, 14, ...]. (sequence A053002 in the OEIS)

Nielsen, Mikkel Slot. (July 2016). Undergraduate convexity : problems and solutions. p. 162. ISBN 9789813146211. OCLC 951172848.
Todd, John (1975). "The lemniscate constants". ACM DL.

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