A094642 Decimal expansion of log(Pi/2).
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, 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, 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
Offset: 0
Examples
log(Pi/2) = 0.45158270528945486472619522989488214357179467855505...
References
- George Boros and Victor Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge, 2004, Chap. 7.
- Jonathan Borwein and Peter Borwein, Pi and the AGM, John Wiley & Sons, New York, 1987, Chap. 11.
- Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, pp. 43-44.
Links
- Dirk Huylebrouck, Similarities in irrationality proofs for Pi, ln2, zeta(2) and zeta(3), Amer. Math. Monthly, Vol. 108, No. 3 (2001), pp. 222-231.
- Jonathan Sondow, A faster product for pi and a new integral for ln(pi/2), The American Mathematical Monthly, Vol. 112, No. 8 (2005), pp. 729-734; Editor's endnotes, ibid., Vol. 113, No. 7 (2006), pp. 670-671; arXiv preprint, arXiv:math/0401406 [math.NT], 2004.
Programs
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Mathematica
RealDigits[ Log[Pi/2], 10, 111][[1]]
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PARI
log(Pi/2) \\ Charles R Greathouse IV, Jun 23 2014
Formula
Equals Sum_{n>=1} zeta(2*n)/(n*2^(2*n)) (cf. Boros & Moll p. 131). - Jean-François Alcover, Apr 29 2013
Equals Re(log(log(I))). - Stanislav Sykora, May 09 2015
Equals Integral_{-oo..+oo} -log(1/2 + i*z)/cosh(Pi*z) dz, where i is the imaginary unit. - Peter Luschny, Apr 08 2018
Equals Integral_{0..Pi/2} (2/(Pi-2*t)-tan(t)) dt. - Clark Kimberling, Jul 10 2020
Equals -Sum_{k>=1} log(1 - 1/(2*k)^2). - Amiram Eldar, Aug 12 2020
Equals Sum_{k>=1} (-1)^(k+1) * log(1 + 1/k). - Amiram Eldar, Jun 26 2021