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.
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
cons,nonn
AUTHOR
Robert G. Wilson v, Aug 03 2002
STATUS
approved