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%I #53 Nov 03 2024 09:32:43
%S 4,5,1,5,8,2,7,0,5,2,8,9,4,5,4,8,6,4,7,2,6,1,9,5,2,2,9,8,9,4,8,8,2,1,
%T 4,3,5,7,1,7,9,4,6,7,8,5,5,5,0,5,6,3,1,7,3,9,2,9,4,3,0,6,1,9,7,8,7,4,
%U 4,1,4,7,9,1,5,1,3,1,3,6,4,1,7,7,7,5,9,9,4,3,2,7,9,0,7,1,0,2,0,1,6,0,0,0,8
%N Decimal expansion of log(Pi/2).
%D George Boros and Victor Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge, 2004, Chap. 7.
%D Jonathan Borwein and Peter Borwein, Pi and the AGM, John Wiley & Sons, New York, 1987, Chap. 11.
%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, pp. 43-44.
%H Dirk Huylebrouck, <a href="https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Huylebrouck222-231.pdf">Similarities in irrationality proofs for Pi, ln2, zeta(2) and zeta(3)</a>, Amer. Math. Monthly, Vol. 108, No. 3 (2001), pp. 222-231.
%H Jonathan Sondow, <a href="https://www.jstor.org/stable/30037575">A faster product for pi and a new integral for ln(pi/2)</a>, The American Mathematical Monthly, Vol. 112, No. 8 (2005), pp. 729-734; <a href="http://www.jstor.org/stable/27642026">Editor's endnotes</a>, ibid., Vol. 113, No. 7 (2006), pp. 670-671; <a href="https://arxiv.org/abs/math/0401406">arXiv preprint</a>, arXiv:math/0401406 [math.NT], 2004.
%F Equals Sum_{n>=1} zeta(2*n)/(n*2^(2*n)) (cf. Boros & Moll p. 131). - _Jean-François Alcover_, Apr 29 2013
%F Equals Re(log(log(I))). - _Stanislav Sykora_, May 09 2015
%F Equals Integral_{-oo..+oo} -log(1/2 + i*z)/cosh(Pi*z) dz, where i is the imaginary unit. - _Peter Luschny_, Apr 08 2018
%F Equals Integral_{0..Pi/2} (2/(Pi-2*t)-tan(t)) dt. - _Clark Kimberling_, Jul 10 2020
%F Equals -Sum_{k>=1} log(1 - 1/(2*k)^2). - _Amiram Eldar_, Aug 12 2020
%F Equals Sum_{k>=1} (-1)^(k+1) * log(1 + 1/k). - _Amiram Eldar_, Jun 26 2021
%F Equals A053510 - A002162. - _R. J. Mathar_, Jun 15 2023
%e log(Pi/2) = 0.45158270528945486472619522989488214357179467855505...
%t RealDigits[ Log[Pi/2], 10, 111][[1]]
%o (PARI) log(Pi/2) \\ _Charles R Greathouse IV_, Jun 23 2014
%Y Cf. A019669, A094643.
%K cons,easy,nonn
%O 0,1
%A _Jonathan Sondow_ and _Robert G. Wilson v_, May 18 2004