OFFSET
1,1
COMMENTS
Nesterenko proves that this constant is transcendental (he cites Chudnovsky as the first to show this); in fact it is algebraically independent of Pi and exp(sqrt(3)*Pi) over Q. - Charles R Greathouse IV, Nov 11 2013
REFERENCES
H. B. Dwight, Tables of Integrals and other Mathematical Data. 860.18, 860.19 in Definite Integrals. New York, U.S.A.: Macmillan Publishing, 1961, p. 230.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 33.
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..1000
Alessandro Languasco and Pieter Moree, Euler constants from primes in arithmetic progression, arXiv:2406.16547 [math.NT], 2024. See p. 8.
Yu V. Nesterenko, Modular functions and transcendence questions, Sbornik: Mathematics 187:9 (1996), pp. 1319-1348. (English translation)
Andrea Pinos, Gamma of reciprocal by Laplace.
Simon Plouffe, GAMMA(1/3).
FORMULA
From Amiram Eldar, Jun 25 2021: (Start)
Equals 2^(7/9) * Pi^(1/3) * K((sqrt(3)-1)/(2*sqrt(2)))^(1/3)/3^(1/12), where K is the complete elliptic integral of the first kind.
Equals 2^(7/9) * Pi^(2/3) /(AGM(2, sqrt(2+sqrt(3)))^(1/3) * 3^(1/12)), where AGM is the arithmetic-geometric mean. (End)
From Andrea Pinos, Aug 12 2023: (Start)
Equals Integral_{x=0..oo} 3*exp(-(x^3)) dx = 3*A202623.
General result: Gamma(1/n) = Integral_{x=0..oo} n*exp(-(x^n)) dx. (End)
EXAMPLE
Gamma(1/3) = 2.6789385347077476336556929409746776441286893779573011009...
MATHEMATICA
RealDigits[ N[ Gamma[1/3], 110]][[1]]
PROG
(PARI) default(realprecision, 1080); x=gamma(1/3); for (n=1, 1000, d=floor(x); x=(x-d)*10; write("b073005.txt", n, " ", d)); \\ Harry J. Smith, Apr 19 2009
(Magma) R:= RealField(100); SetDefaultRealField(R); Gamma(1/3); // G. C. Greubel, Mar 10 2018
CROSSREFS
KEYWORD
AUTHOR
Robert G. Wilson v, Aug 03 2002
STATUS
approved