The reciprocal of the arithmetic-geometric mean of 1 and 
,
(OEIS A014549), where 
is the lemniscate constant,
is the complete
elliptic integral of the first kind, 
is a Jacobi
theta function, 
is the gamma function, and 
, 
, 
are Carlson elliptic
integrals. This correspondence was first noticed by Gauss, and was the basis
for his exploration of the lemniscate function
(Borwein and Bailey 2003, pp. 13-15).
Two rapidly converging series for 
are given by
(Finch 2003, p. 421).
Gauss's constant has continued fraction [0,
1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, ...] (OEIS A053002).
The inverse of Gauss's constant is given by
 |
(15)
|
(OEIS A053004; Finch 2003, p. 420; Borwein and Bailey 2003, p. 13), which has [1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1,
15, 1, ...] (OEIS A053003).
The value
 |
(16)
|
(OEIS A097057) is sometimes called the ubiquitous constant (Spanier and Oldham 1987; Schroeder 1994; Finch 2003, p. 421),
and
 |
(17)
|
(OEIS A076390) is sometimes called the second
lemniscate constant (Finch 2003, p. 421).
Gauss's constants 
and 
are related to the lemniscate constant 
by
(Finch 2003, p. 420).
See also
Arithmetic-Geometric Mean,
Gauss-Kuzmin-Wirsing Constant,
Lemniscate Constant,
Lemniscate
Function,
Pythagoras's Constant
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References
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.
New York: Wiley, p. 5, 1987.Borwein, J. and Bailey, D. Mathematics
by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A
K Peters, 2003.Goldman, J. R. The
Queen of Mathematics: An Historically Motivated Guide to Number Theory. Wellesley,
MA: A K Peters, p. 92, 1997.Finch, S. R. Mathematical
Constants. Cambridge, England: Cambridge University Press, 2003.Gosper,
R. W. "A Calculus of Series Rearrangements." In Algorithms
and Complexity: New Directions and Recent Results. Proc. 1976 Carnegie-Mellon Conference
(Ed. J. F. Traub). New York: Academic Press, pp. 121-151, 1976.Lewanowicz,
S. and Paszowski, S. "An Analytic Method for Convergence Acceleration of Certain
Hypergeometric Series." Math. Comput. 64, 691-713, 1995.Schroeder,
M. "How Probable is Fermat's Last Theorem?" Math. Intell. 16,
19-20, 1994.Sloane, N. J. A. Sequences A014549,
A053002, A053003,
A053004, A076390,
and A097057 in "The On-Line Encyclopedia
of Integer Sequences."Spanier, J. and Oldham, K. B. "The
Kelvin Functions." Ch. 55 in An
Atlas of Functions. Washington, DC: Hemisphere, 1987.Todd, J.
"The Lemniscate Constant." Comm. ACM 18, 14-19 and 462, 1975.Referenced
on Wolfram|Alpha
Gauss's Constant
Cite this as:
Weisstein, Eric W. "Gauss's Constant."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GausssConstant.html
Subject classifications
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