A068467 - OEIS

%I #45 Sep 08 2022 08:45:05

%S 9,0,6,4,0,2,4,7,7,0,5,5,4,7,7,0,7,7,9,8,2,6,7,1,2,8,8,9,6,6,9,1,8,0,

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

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

%N Decimal expansion of (1/4)! = Gamma(5/4).

%H G. C. Greubel, <a href="/A068467/b068467.txt">Table of n, a(n) for n = 0..10000</a>

%H J. M. Borwein and I. J. Zucker, <a href="http://imajna.oxfordjournals.org/content/12/4/519.abstract">Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind</a>, IMA Journal of Numerical Analysis, vol. 12, no. 4, pp. 519-526, 1992.

%H Greg Martin, <a href="http://arxiv.org/abs/0907.4348">A product of Gamma function values at fractions with the same denominator</a>, arXiv:0907.4384v1 [math.CA], 24-July-2009

%H Albert Nijenhuis, <a href="http://arxiv.org/abs/0907.1689">Small Gamma Products with Simple Values</a>, arXiv:0907.1689v1 [math.CA], 9-July-2009.

%H Raimundas Vidunas, <a href="http://arxiv.org/abs/math/0403510">Expressions for values of the gamma function</a>, arXiv:math/0403510 [math.CA], 30-March-2004.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Particular_values_of_the_gamma_function">Particular values of the Gamma function</a>

%H <a href="/index/Ga#gamma_function">Index to sequences related to the Gamma function</a>

%F 2^(3/4)*(2/e^(16*Pi) + 1)* Pi^(3/4)/(2^(13/16)/(sqrt(2) - 1)^(1/4) + 2^(1/4) + 1) is a very good approximation (~88 digits) which becomes exact if you replace (2/e^(16*Pi) + 1) by EllipticTheta[3,0,exp(-(16*Pi))]. [R. W. Gosper, Posting to Math Fun Mailing List, Dec 27 2011.]

%F Equals A068466 /4 . - _R. J. Mathar_, Jan 10 2013

%F Also equals integral_{0..oo} exp(-x^4) dx. - _Jean-François Alcover_, Mar 29 2013

%F Equals 2^(-5/4)*Pi^(3/4)*Product_{k>=1} tanh(Pi*k/2). - _Keshav Raghavan_, Aug 25 2016

%e 0.906402477055477077982671288966918000748791920720...

%p evalf(GAMMA(5/4)) ; # _R. J. Mathar_, Jan 10 2013

%t RealDigits[Gamma[5/4],10,120][[1]] (* _Harvey P. Dale_, Aug 23 2013 *)

%o (PARI) gamma(5/4) \\ _Altug Alkan_, Sep 18 2016

%o (Magma) SetDefaultRealField(RealField(100)); Gamma(5/4); // _G. C. Greubel_, Mar 11 2018

%Y Cf. A202623.

%K cons,easy,nonn

%O 0,1

%A _Benoit Cloitre_, Mar 10 2002

%E Removed leading zero and adjusted offset, _R. J. Mathar_, Feb 06 2009

%E Additional reference from _Joerg Arndt_, Dec 28 2011

%E Edited by _N. J. A. Sloane_, Dec 28 2011