A073005 Decimal expansion of Gamma(1/3).

2, 6, 7, 8, 9, 3, 8, 5, 3, 4, 7, 0, 7, 7, 4, 7, 6, 3, 3, 6, 5, 5, 6, 9, 2, 9, 4, 0, 9, 7, 4, 6, 7, 7, 6, 4, 4, 1, 2, 8, 6, 8, 9, 3, 7, 7, 9, 5, 7, 3, 0, 1, 1, 0, 0, 9, 5, 0, 4, 2, 8, 3, 2, 7, 5, 9, 0, 4, 1, 7, 6, 1, 0, 1, 6, 7, 7, 4, 3, 8, 1, 9, 5, 4, 0, 9, 8, 2, 8, 8, 9, 0, 4, 1, 1, 8, 8, 7, 8, 9, 4, 1, 9, 1, 5

Offset: 1

Author

Robert G. Wilson v, Aug 03 2002

Keywords

• cons
• nonn

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

Examples

			Gamma(1/3) = 2.6789385347077476336556929409746776441286893779573011009...
		

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.
• Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 43, equation 43:4:8 at page 413.

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).
• Wikipedia, Particular values of the Gamma function: General rational arguments.
• Index entries for transcendental numbers.
• Index to sequences related to the Gamma function.

Crossrefs

Cf. A073006, A202623, A203145, A256165.

Programs

Magma

R:= RealField(100); SetDefaultRealField(R); Gamma(1/3); // G. C. Greubel, Mar 10 2018

Mathematica

RealDigits[ N[ Gamma[1/3], 110]][[1]]

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

Formula

this * A073006 = A186706. - R. J. Mathar, Jan 15 2021

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)

Equals 3*A202623 = exp(A256165). - Hugo Pfoertner, Jun 28 2024

Equals (2^(1/3)*Pi*C*3^(1/2))^(1/3), where C = A118292 = Integral {0..1} 2/sqrt(1-x^3) is the transcendental butterfly constant. - Jan Lügering, Feb 08 2025