A173623 Decimal expansion of Pi*log(2)/2.
1, 0, 8, 8, 7, 9, 3, 0, 4, 5, 1, 5, 1, 8, 0, 1, 0, 6, 5, 2, 5, 0, 3, 4, 4, 4, 4, 9, 1, 1, 8, 8, 0, 6, 9, 7, 3, 6, 6, 9, 2, 9, 1, 8, 5, 0, 1, 8, 4, 6, 4, 3, 1, 4, 7, 1, 6, 2, 8, 9, 7, 6, 2, 6, 5, 9, 7, 1, 5, 4, 2, 7, 4, 5, 8, 8, 3, 7, 0, 9, 9, 3, 2, 1, 5, 1, 6, 4, 4, 8, 0, 8, 0, 5, 3, 3, 1, 5, 1, 2, 5, 2, 8, 8, 0
Offset: 1
Examples
1.08879304515180106525034444...
References
- Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4.3, p. 22.
- Paul J. Nahin, Inside Interesting Integrals, Springer 2015, ISBN 978-1493912766.
Links
- Su Hu and Min-Soo Kim, Euler's integral, multiple cosine function and zeta values, arXiv:2201.01124 [math.NT], 2022.
- K. S. Kölbig, On the integral int_0^Pi/2 log^n cos x log^p sin x dx, Math. Comp. 40 (162) (1983) 565-570, r_{1,0}.
- Richard J. Mathar, Chebyshev approximation of x^m(-log x)^l in the interval 0 <= x <= 1, arXiv:2408.15212 [math.CA], 2024. See p. 2.
- Kazuhiro Onodera, Generalized log sine integrals and the Mordell-Tornheim zeta values, Trans. Am. Math. Soc. 363 (3) (2010), 1463-1485.
Programs
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Maple
Pi/2*log(2) ; evalf(%) ;
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Mathematica
RealDigits[Pi*Log[2]/2, 10, 100][[1]] (* Amiram Eldar, Jul 13 2020 *)
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PARI
Pi*log(2)/2 \\ Stefano Spezia, Oct 21 2024
Formula
Equals abs(Integral_{x=0..Pi/2} log(sin(x)) dx).
Equals A086054 / 2.
From Amiram Eldar, Jul 13 2020: (Start)
Equals Integral_{x=0..1} arcsin(x)/x dx.
Equals Integral_{x=0..Pi/2} x*cot(x) dx. (End)
Equals Integral_{x = 0..1} log(x + 1/x)/(1 + x^2) dx (Nahin, 2.4.4) = (1/2)*Integral_{x = 0..oo} log(x^2 + 4)/(x^2 + 4) dx = (1/2)*Integral_{x = 0..oo} log(x^2 + 1)/(x^2 + 1) dx = Integral_{x = 0..oo} log(x^2 + 64)/(x^2 + 64) dx. - Peter Bala, Jul 22 2022
Equals 3F2(1/2,1/2,1/2 ; 3/2,3/2 ; 1). - R. J. Mathar, Aug 19 2024