A193712 Decimal expansion of Pi*zeta(3)/4.

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

Offset: 0

Author

Seiichi Kirikami, Aug 31 2011

Keywords

• nonn
• cons

Comments

The absolute value of Integral_{x=0..Pi/2} x^2*log(2*cos(x)) dx.

The absolute value of (d/db(d^2/da^2(Integral_{x=0..Pi/2} cos(ax)*(2*cos(x))^b dx))).

The absolute value of (Pi/2)*(d/db(d^2/da^2(gamma(b+1)/gamma((b+a)/2+1)/gamma((b-a)/2+1))) at a=0 and b=0. - Seiichi Kirikami and Peter J. C. Moses

Examples

			0.94409328404076973180...
		

References

• I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 3.631.9

Links

• Masato Kobayashi, Integral representations for zeta(3) with the inverse sine function, arXiv:2108.01247 [math.NT], 2021.

Crossrefs

Cf. A000796, A002117, A152584, A193713, A194655.

Programs

Mathematica

RealDigits[ N[Pi Zeta[3]/4, 150]][[1]]

Formula

Equals A000796*A002117/4.

Equals 2 * Integral_{x=0..1} arcsin(x)^2*arccos(x)/x dx (Kobayashi, 2021). - Amiram Eldar, Jun 23 2023