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A115252 Decimal expansion of -(Pi*log((sqrt(2*Pi)*Gamma(3/4))/Gamma(1/4)))/2. 7
2, 6, 0, 4, 4, 2, 8, 0, 6, 3, 0, 0, 9, 8, 8, 4, 4, 5, 5, 4, 0, 0, 9, 3, 8, 6, 8, 7, 8, 9, 7, 2, 7, 2, 1, 9, 5, 3, 1, 8, 1, 9, 1, 7, 7, 7, 2, 3, 1, 4, 2, 6, 7, 4, 9, 8, 7, 6, 8, 7, 7, 9, 2, 1, 0, 5, 7, 7, 1, 6, 0, 3, 8, 1, 4, 7, 3, 1, 7, 3, 9, 2, 6, 9, 8, 9, 3, 3, 2, 0, 8, 0, 4, 0, 0, 9, 1, 4, 9, 8, 1, 1, 7, 1, 3 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
This sequence (its negated version) is also the decimal expansion of the first Malmsten integral int_{x=1..infinity} log(log(x))/(1 + x^2) dx = int_{x=0..1} log(log(1/x))/(1 + x^2) dx = int_{x=0..infinity} 0.5*log(x)/cosh(x) dx = int_{x=Pi/4..Pi/2} log(log(tan(x))) dx = (1/2)*Pi*log(2) + (3/4)*Pi*log(Pi) - Pi*log(Gamma(1/4)). - Iaroslav V. Blagouchine, Mar 29 2015
LINKS
I. V. Blagouchine, Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal, Volume 35, Issue 1, pp. 21-110, 2014, DOI: 10.1007/s11139-013-9528-5. PDF file
Eric Weisstein's World of Mathematics, Vardi's Integral
FORMULA
Equals integral_[0..1] log(1/log(1/x))/(1+x^2) dx. - Jean-François Alcover, Jan 28 2015
EXAMPLE
0.26044280630098844554009386878972721953181917772314...
MATHEMATICA
RealDigits[-Pi/2*Log[Sqrt[2 Pi] Gamma[3/4]/Gamma[1/4]], 10, 111][[1]] (* Robert G. Wilson v, Dec 06 2014 *)
PROG
(PARI) (-Pi*log((sqrt(2*Pi)*gamma(3/4))/gamma(1/4)))/2 \\ Michel Marcus, Dec 06 2014
CROSSREFS
Cf. A256127 (second Malmsten integral), A256128 (third Malmsten integral), A256129 (fourth Malmsten integral), A068466 (Gamma(1/4)), A256166 (log(Gamma(1/4))), A002162 (log 2), A053510 (log Pi).
Sequence in context: A021388 A011040 A274402 * A108431 A190144 A019967
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Jan 17 2006
STATUS
approved

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Last modified March 29 10:22 EDT 2024. Contains 371268 sequences. (Running on oeis4.)