A225119 Decimal expansion of Integral_{x=0..Pi/2} sin(x)^(3/2) dx.

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

Offset: 0

Author

Jean-François Alcover, Apr 29 2013

Keywords

• nonn
• cons
• easy

Examples

			0.87401918476403993682161319663037313789425165047720772093894...
		

References

• George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 195.
• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constant p. 102 and Section 6.1 Gauss' Lemniscate Constant p. 422.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Index entries for transcendental numbers.

Crossrefs

Cf. A062539, A068466, A093341, A085565.

Programs

Maple

evalf((1/3)*sqrt(2)*EllipticK(1/sqrt(2)), 120); # Vaclav Kotesovec, Apr 22 2015

Mathematica

RealDigits[1/3*Sqrt[2]*EllipticK[1/2], 10, 100][[1]]

PARI

sqrt(Pi)*gamma(1/4)/(6*gamma(3/4)) \\ G. C. Greubel, Apr 01 2017

PARI

ellK(sqrt(1/2))*sqrt(2)/3 \\ Charles R Greathouse IV, Feb 04 2025

Formula

Equals 1/3 * sqrt(2) * ellipticK(1/2), (defined as in Mathematica).

Equals sqrt(2)/6 * Pi * hypergeom([1/2,1/2],[1],1/2).

Equals gamma(1/4)^2/(6*sqrt(2*Pi)).

Equals sqrt(Pi)*gamma(1/4)/(6*gamma(3/4)).

Equals Integral_{0..1} (1-x^2)^(1/4) dx.

Equals Integral_{0..1} sqrt(1-x^4) dx. - Charles R Greathouse IV, Aug 21 2017

Equals (2/3)*A085565. - Peter Bala, Oct 27 2019

Equals A062539/3. - Hugo Pfoertner, Dec 15 2024