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A115252 Decimal expansion of -(Pi*log((sqrt(2*Pi)*Gamma(3/4))/Gamma(1/4)))/2. 6

%I

%S 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,

%T 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,

%U 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

%N Decimal expansion of -(Pi*log((sqrt(2*Pi)*Gamma(3/4))/Gamma(1/4)))/2.

%C 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

%H G. C. Greubel, <a href="/A115252/b115252.txt">Table of n, a(n) for n = 0..5000</a>

%H I. V. Blagouchine, <a href="http://dx.doi.org/10.1007/s11139-013-9528-5">Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results</a>, The Ramanujan Journal, Volume 35, Issue 1, pp. 21-110, 2014, DOI: 10.1007/s11139-013-9528-5. <a href="http://www.researchgate.net/publication/257381156_Rediscovery_of_Malmsten%27s_integrals_their_evaluation_by_contour_integration_methods_and_some_related_results">PDF file</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VardisIntegral.html">Vardi's Integral</a>

%F Equals integral_[0..1] log(1/log(1/x))/(1+x^2) dx. - _Jean-Fran├žois Alcover_, Jan 28 2015

%e 0.26044280630098844554009386878972721953181917772314...

%t RealDigits[-Pi/2*Log[Sqrt[2 Pi] Gamma[3/4]/Gamma[1/4]], 10, 111][[1]] (* _Robert G. Wilson v_, Dec 06 2014 *)

%o (PARI) (-Pi*log((sqrt(2*Pi)*gamma(3/4))/gamma(1/4)))/2 \\ _Michel Marcus_, Dec 06 2014

%Y 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).

%K nonn,cons

%O 0,1

%A _Eric W. Weisstein_, Jan 17 2006

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Last modified February 23 05:58 EST 2019. Contains 320411 sequences. (Running on oeis4.)