%I #36 Oct 29 2021 16:12:25

%S 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,

%T 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,

%U 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

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

%D George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 195.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constant, p. 102.

%H G. C. Greubel, <a href="/A225119/b225119.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

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

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

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

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

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

%F Equals Integral_{0..1} sqrt(1-x^4) dx. - _Charles R Greathouse IV_, Aug 21 2017

%F Equals (2/3)*A085565. - _Peter Bala_, Oct 27 2019

%e 0.87401918476403993682161319663037313789425165047720772093894...

%p evalf((1/3)*sqrt(2)*EllipticK(1/sqrt(2)), 120); # _Vaclav Kotesovec_, Apr 22 2015

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

%o (PARI) sqrt(Pi)*gamma(1/4)/(6*gamma(3/4)) \\ _G. C. Greubel_, Apr 01 2017

%Y Cf. A068466, A093341, A085565.

%K nonn,cons,easy

%O 0,1

%A _Jean-François Alcover_, Apr 29 2013