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

1,3

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

Table of n, a(n) for n=1..105.

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.

Formula

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

Equals A086054 / 2.

From Amiram Eldar, Jul 13 2020: (Start)

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

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

Example

1.08879304515180106525034444...

Maple

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

Mathematica

RealDigits[Pi*Log[2]/2, 10, 100][[1]] (* Amiram Eldar, Jul 13 2020 *)

Prog

(PARI) Pi*log(2)/2 \\ Stefano Spezia, Oct 21 2024

Crossrefs

Cf. A000984, A086054, A164583.

Sequence in context: A127196 A350715 A294795 * A070481 A011109 A248622

Adjacent sequences: A173620 A173621 A173622 * A173624 A173625 A173626

Keyword

nonn,cons

Author

R. J. Mathar, Nov 08 2010

Status

approved