A110544 Decimal expansion of -Integral {x=1..2} log gamma(x) dx.

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

Offset

0,2

Links

Table of n, a(n) for n=0..105.

Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (1.8.1)

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

Eric Weisstein's World of Mathematics, Gamma Function.

Formula

Equals zeta'(0)+1 = -1/2*log(2*Pi)+1. - Jean-François Alcover, Jun 10 2013

From Amiram Eldar, Jul 05 2020: (Start)

Equals Sum_{k>=2} (1/(k + 1) - 1/(2*k))*(zeta(k)-1).

Equals Integral_{x=0..1} (1/2 - 1/(1 - x) - 1/log(x)) dx/log(x). (End)

Equals -Integral_{x=1..oo} ({x}-1/2)/x dx, where {.} is the fractional part [Nahin]. - R. J. Mathar, May 16 2024

Equals 1 - A075700 = log(A229495). - Hugo Pfoertner, Sep 05 2024

Example

0.081061466795327258219670263594382360138602526362216587182848459...

Mathematica

RealDigits[ N[ -Integrate[ Log[ Gamma[ x]], {x, 1, 2}], 128], 10, 128]
RealDigits[ 1/2*Log[2*Pi]-1, 10, 105] // First // Prepend[#, 0]& (* Jean-François Alcover, Jun 10 2013 *)

Prog

(PARI) -intnum(x=1, 2, log(gamma(x))) \\ Michel Marcus, Jul 05 2020

Crossrefs

Cf. A075700, A110543, A229495.

Sequence in context: A373508 A194732 A217739 * A320084 A153855 A340065

Adjacent sequences: A110541 A110542 A110543 * A110545 A110546 A110547

Keyword

cons,nonn

Author

Robert G. Wilson v, Jul 25 2005

Status

approved