A075700 Decimal expansion of -zeta'(0).

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

Offset

0,1

Comments

The probability density function for the standard normal distribution is e^(-x^2/2 + zeta'(0)). - Rick L. Shepherd, Mar 08 2014

For every x > 0, PolyGamma(-2, x+1) - (PolyGamma(-2, x) + x*log(x) - x) equals this constant -zeta'(0), where polygamma functions of negative indices are defined for x > 0 as: PolyGamma(-1, x) = log(Gamma(x)), PolyGamma(-(n+1), x) = Integral_{t=0..x} PolyGamma(-n, x) dx, n >= 1. - Jianing Song, Apr 20 2021

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

J. Sondow and E. W. Weisstein, MathWorld: Wallis Formula.

Eric Weisstein's World of Mathematics, Log Gamma Function.

Eric Weisstein's World of Mathematics, Stirling's Approximation.

Wikipedia, Gamma function.

Wikipedia, Normal curve

Index entries for zeta function.

Formula

Equals log(2*Pi)/2 = A061444/2 = log(A019727).

Equals Integral_{x=0..1} log(Gamma(x)) dx. - Jean-François Alcover, Apr 29 2013

More generally, equals t-t*log(t)+Integral_{x=t..(t+1)} (log(Gamma(x)) dx for any t>=0 (the Raabe formula). - Stanislav Sykora, May 14 2015

Equals lim_{k->oo} log(k!) + k - (k + 1/2)*log(k) (by Stirling's formula). - Amiram Eldar, Aug 21 2020

Example

0.91893853320467274178032...

Maple

evalf(log(2*Pi)/2, 120); # Muniru A Asiru, Oct 08 2018

Mathematica

Log[Sqrt[2*Pi]] // RealDigits[#, 10, 104] & // First (* Jean-François Alcover, Apr 29 2013 *)

Prog

(PARI) -zeta'(0) \\ Charles R Greathouse IV, Mar 28 2012
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Log(2*Pi(R))/2; // G. C. Greubel, Oct 07 2018

Crossrefs

Cf. A019727, A061444, A257549.

Sequence in context: A299622 A163899 A198758 * A021843 A362752 A231931

Adjacent sequences: A075697 A075698 A075699 * A075701 A075702 A075703

Keyword

cons,nonn

Author

Benoit Cloitre, Oct 02 2002

Extensions

Normalized representation (leading zero and offset) R. J. Mathar, Jan 25 2009

Status

approved