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

%I A225119 #43 Feb 04 2025 22:55:34
%S A225119 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 A225119 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 A225119 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 A225119 Decimal expansion of Integral_{x=0..Pi/2} sin(x)^(3/2) dx.
%D A225119 George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 195.
%D A225119 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.
%H A225119 G. C. Greubel, <a href="/A225119/b225119.txt">Table of n, a(n) for n = 0..10000</a>
%H A225119 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A225119 Equals 1/3 * sqrt(2) * ellipticK(1/2), (defined as in Mathematica).
%F A225119 Equals sqrt(2)/6 * Pi * hypergeom([1/2,1/2],[1],1/2).
%F A225119 Equals gamma(1/4)^2/(6*sqrt(2*Pi)).
%F A225119 Equals sqrt(Pi)*gamma(1/4)/(6*gamma(3/4)).
%F A225119 Equals Integral_{0..1} (1-x^2)^(1/4) dx.
%F A225119 Equals Integral_{0..1} sqrt(1-x^4) dx. - _Charles R Greathouse IV_, Aug 21 2017
%F A225119 Equals (2/3)*A085565. - _Peter Bala_, Oct 27 2019
%F A225119 Equals A062539/3. - _Hugo Pfoertner_, Dec 15 2024
%e A225119 0.87401918476403993682161319663037313789425165047720772093894...
%p A225119 evalf((1/3)*sqrt(2)*EllipticK(1/sqrt(2)), 120); # _Vaclav Kotesovec_, Apr 22 2015
%t A225119 RealDigits[1/3*Sqrt[2]*EllipticK[1/2], 10, 100][[1]]
%o A225119 (PARI) sqrt(Pi)*gamma(1/4)/(6*gamma(3/4)) \\ _G. C. Greubel_, Apr 01 2017
%o A225119 (PARI) ellK(sqrt(1/2))*sqrt(2)/3 \\ _Charles R Greathouse IV_, Feb 04 2025
%Y A225119 Cf. A062539, A068466, A093341, A085565.
%K A225119 nonn,cons,easy
%O A225119 0,1
%A A225119 _Jean-François Alcover_, Apr 29 2013