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:

where agm denotes the arithmetic-geometric mean. The constant is named after Carl Friedrich Gauss, who on May 30, 1799 discovered that

so that

where β denotes the beta function.

Table of contents

1 Uses 1.1 Lemniscate constants 1.2 Calculation 2 References 3 See also 3.3 External links

Uses

Gauss's constant may be used as a closed-form expression for the Gamma function at argument 1/4:

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

Lemniscate constants

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

and the second constant:

which arise in finding the arc length of a lemniscate.

Calculation

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

as well as the rapidly converging series:

Gauss's constant has continued fraction [0, 1, 5, 21, 3, 4, 14...]

References

• Eric W. Weisstein. "Gauss's Constant." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/GausssConstant.html
• Sequences A014549 and A053002 in OEIS

See also

• Arithmetic-geometric mean
• Theta function
• Beta function
• Gamma function

External links

• Gauss's Constant -- from MathWorld

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