Gauss's constant

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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:

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

so that

where Β denotes the beta function.

Relations to other constants[edit]

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

Alternatively,

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

Lemniscate constants[edit]

Gauss's constant may be used in the definition of the lemniscate constants.

Gauss and others[2][3] use the equivalent of

which is the lemniscate constant, prominent in the theory of lemniscatic functions.

However, John Todd[4] uses a different terminology – in his article, the numbers and are called the lemniscate constants, the first of which is

and the second constant:

They arise in finding the arc length of a lemniscate of Bernoulli. and were proven transcendental by Theodor Schneider in 1937 and 1941, respectively.[4]

Other formulas[edit]

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

as well as the rapidly converging series

The constant is also given by the infinite product

It appears in the evaluation of the integrals

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

See also[edit]

References[edit]

  1. ^ Nielsen, Mikkel Slot. (July 2016). Undergraduate convexity : problems and solutions. p. 162. ISBN 9789813146211. OCLC 951172848.
  2. ^ Kobayashi, Hiroyuki; Takeuchi, Shingo (2019), Applications of generalized trigonometric functions with two parameters, arXiv:1903.07407
  3. ^ Asai, Tetsuya (2007), Elliptic Gauss Sums and Hecke L-values at s=1, arXiv:0707.3711
  4. ^ a b Todd, John (1975). "The lemniscate constants". ACM DL.