A196878 -id:A196878 - cp's OEIS frontend

A196877 Decimal expansion of Pi/2*(Pi^2/12 + (log(2))^2).

2, 0, 4, 6, 6, 2, 2, 0, 2, 4, 4, 7, 2, 7, 4, 0, 6, 4, 6, 1, 6, 9, 6, 4, 1, 0, 0, 8, 1, 7, 6, 9, 7, 3, 4, 7, 6, 6, 3, 7, 4, 4, 1, 9, 5, 3, 4, 9, 4, 6, 5, 6, 2, 6, 0, 6, 1, 0, 2, 6, 8, 5, 5, 2, 7, 2, 5, 9, 0, 6, 6, 8, 7, 9, 5, 1, 2, 1, 7, 3, 3, 6, 5, 8, 4, 6, 8, 8, 4, 6, 7, 6, 3, 2, 9, 1, 2, 5, 2, 5, 3, 4, 3, 4, 7

Offset: 1

Seiichi Kirikami, Oct 07 2011

The value of the integral_{x=0..Pi/2} log(sin(x))^2 dx. The value of sqrt(Pi)/2*(d^2/da^2(gamma((a+1)/2)/gamma(a/2+1))) at a=0. - Seiichi Kirikami and Peter J. C. Moses, Oct 07 2011

			2.04662202447274064616964100817...

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 3.621.1

G. C. Greubel, Table of n, a(n) for n = 1..10000
K. S. Kolbig, On the integral int_0^Pi/2 log^n cos x log^p sin x dx, Math. Comp. 40 (162) (1983) 565-570, r_{2,0}

Cf. A002162, A019669, A072691, A173623, A196878.

RealDigits[N[Pi/2 (Pi^2/12 + Log[2]^2),150]][[1]]

Pi/2*(Pi^2/12+(log(2))^2) \\ Michel Marcus, Jan 13 2015

Equals A019669*(A072691 + A002162^2).

Equals Integral_{x=0..1} log(x)^2/sqrt(1-x^2) dx. - Amiram Eldar, May 27 2023

A217708 Decimal expansion of integral_{0, Pi/4} log(sin(x))^3 dx.

Original entry on oeis.org

6, 0, 3, 7, 5, 8, 1, 1, 0, 6, 3, 0, 9, 5, 9, 8, 1, 7, 0, 2, 6, 7, 4, 3, 6, 8, 9, 2, 6, 5, 0, 2, 6, 7, 4, 6, 7, 5, 5, 0, 3, 9, 8, 2, 8, 5, 8, 8, 8, 6, 8, 6, 2, 7, 1, 7, 5, 1, 8, 0, 4, 9, 0, 1, 3, 5, 5, 8, 7, 4, 2, 2, 1, 2, 7, 0, 1, 6, 0, 6, 7, 4, 0, 9, 7, 6, 9, 3, 5, 9, 7, 6, 6, 7, 5, 9, 9, 6, 8, 2

Offset: 1

Jean-François Alcover, Mar 20 2013

Curiously, this integral from 0 to Pi/4 is far more difficult to evaluate by hand than the same integral from 0 to Pi/2.

			-6.037581106...

Cf. A196878 (the same integral from 0 to Pi/2)

(1/8)*((-Log[2]^2)*(Pi*Log[2] + 3*Catalan) - 4*Sqrt[2]*(6*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2, 3/2}, 1/2] + HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, 1/2]*Log[8])) // RealDigits[#, 10, 100] & // First
RealDigits[Integrate[Log[Sin[x]]^3,{x,0,Pi/4}],10,120][[1]] (* Harvey P. Dale, Jun 08 2021 *)

A375594 Decimal expansion of Pi(Pi^2log(2) + 4log(2)^3 + 6zeta(3))/48.

Original entry on oeis.org

1, 0, 0, 6, 9, 8, 0, 4, 8, 4, 9, 6, 2, 5, 1, 5, 5, 8, 7, 0, 2, 5, 0, 7, 0, 6, 8, 8, 4, 4, 5, 9, 9, 9, 3, 3, 0, 9, 1, 8, 1, 1, 8, 3, 8, 4, 2, 9, 5, 7, 7, 3, 6, 5, 1, 2, 0, 9, 7, 6, 6, 8, 3, 5, 8, 7, 6, 6, 7, 3, 8, 3, 7, 5, 7, 7, 5, 9, 5, 9, 6, 9, 3, 4, 0, 0, 7, 8, 4, 7, 1, 0, 9, 8, 0, 4, 3, 6, 1, 5, 8, 5

Offset: 1

R. J. Mathar, Aug 20 2024

Apart from a factor sqrt(Pi)/16 the same as Adamchik's generalized Stirling number [1/2,4].

			1.006980484962515...

V. S. Adamchik, On Stirling numbers and Euler sums, J. Comput. Appl. Math. 79 (1) (1997) 119-130.
R. J. Mathar, Chebyshev approximation of x^m*(-log x)^l in the interval 0<=x<=1, arXiv:2408.15212 [math.CA] (2024).

Cf. A019669 (2F1), A173623 (3F2), A318741 (4F3).

1/48*Pi*(Pi^2*log(2)+4*log(2)^3+6*Zeta(3)) ; evalf(%) ;

First[RealDigits[Pi*(Pi^2*Log[2] + 4*Log[2]^3 + 6*Zeta[3])/48, 10, 100]] (* Paolo Xausa, Aug 23 2024 *)

Equals 5F4(1/2,1/2,1/2,1/2,1/2; 3/2,3/2,3/2,3/2; 1) = Sum_{k>= 0} binomial(2k,k)/[2^(2k)*(2k+1)^4].

Equals A196878/6. - R. J. Mathar, Aug 23 2024

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A196877 Decimal expansion of Pi/2*(Pi^2/12 + (log(2))^2).

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A217708 Decimal expansion of integral_{0, Pi/4} log(sin(x))^3 dx.

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A375594 Decimal expansion of Pi(Pi^2log(2) + 4log(2)^3 + 6zeta(3))/48.

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cp's OEIS Frontend

A196877 Decimal expansion of Pi/2*(Pi^2/12 + (log(2))^2).

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A217708 Decimal expansion of integral_{0, Pi/4} log(sin(x))^3 dx.

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A375594 Decimal expansion of Pi*(Pi^2*log(2) + 4*log(2)^3 + 6*zeta(3))/48.

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A375594 Decimal expansion of Pi(Pi^2log(2) + 4log(2)^3 + 6zeta(3))/48.