OFFSET
0,2
LINKS
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
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
KEYWORD
cons,nonn
AUTHOR
Robert G. Wilson v, Jul 25 2005
STATUS
approved