A173623 - OEIS

%I #38 Oct 21 2024 17:31:14

%S 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,

%T 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,

%U 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

%N Decimal expansion of Pi*log(2)/2.

%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4.3, p. 22.

%D Paul J. Nahin, Inside Interesting Integrals, Springer 2015, ISBN 978-1493912766.

%H Su Hu and Min-Soo Kim, <a href="https://arxiv.org/abs/2201.01124">Euler's integral, multiple cosine function and zeta values</a>, arXiv:2201.01124 [math.NT], 2022.

%H K. S. Kölbig, <a href="https://doi.org/10.1090/S0025-5718-1983-0689472-3">On the integral int_0^Pi/2 log^n cos x log^p sin x dx</a>, Math. Comp. 40 (162) (1983) 565-570, r_{1,0}.

%H Richard J. Mathar, <a href="https://arxiv.org/abs/2408.15212">Chebyshev approximation of x^m(-log x)^l in the interval 0 <= x <= 1</a>, arXiv:2408.15212 [math.CA], 2024. See p. 2.

%H Kazuhiro Onodera, <a href="http://dx.doi.org/10.1090/S0002-9947-2010-05176-1">Generalized log sine integrals and the Mordell-Tornheim zeta values</a>, Trans. Am. Math. Soc. 363 (3) (2010), 1463-1485.

%F Equals abs(Integral_{x=0..Pi/2} log(sin(x)) dx).

%F Equals A086054 / 2.

%F From _Amiram Eldar_, Jul 13 2020: (Start)

%F Equals Sum_{k>=0} binomial(2*k,k)/(4^k*(2*k+1)^2) = Sum_{k>=0} A000984(k)/A164583(k).

%F Equals Integral_{x=0..1} arcsin(x)/x dx.

%F Equals Integral_{x=0..Pi/2} x*cot(x) dx. (End)

%F 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

%F Equals 3F2(1/2,1/2,1/2 ; 3/2,3/2 ; 1). - _R. J. Mathar_, Aug 19 2024

%e 1.08879304515180106525034444...

%p Pi/2*log(2) ; evalf(%) ;

%t RealDigits[Pi*Log[2]/2, 10, 100][[1]] (* _Amiram Eldar_, Jul 13 2020 *)

%o (PARI) Pi*log(2)/2 \\ _Stefano Spezia_, Oct 21 2024

%Y Cf. A000984, A086054, A164583.

%K nonn,cons,changed

%O 1,3

%A _R. J. Mathar_, Nov 08 2010