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%I #16 Sep 06 2024 08:05:54
%S 1,5,9,4,9,3,8,9,4,1,4,8,6,3,0,2,8,8,7,1,1,6,8,1,2,9,4,3,8,9,2,1,5,0,
%T 6,7,7,3,6,6,8,1,3,6,0,0,7,1,6,4,6,9,9,9,0,8,5,7,0,0,0,4,6,0,2,9,6,6,
%U 6,0,3,0,8,4,9,5,2,5,8,4,8,0,0,3,0,6,7
%N Decimal expansion of the hypergeometric series 3F2(3/4,5/4,1; 2,2 ;1).
%H W. N. Bailey, <a href="https://doi.org/10.1017/S2040618500033049">Contiguous hypergeometric functions of the type 3F2</a>, Proc. Glasg. Math. Ass. 2 (1954) 62-65.
%H James D. Evans, <a href="https://doi.org/10.1016/0096-3003(90)90011-Q">Evaluation of four irrational definite sine integrals using residue theory</a>, Appl. Math. Comp 36 (1990) 163-172, eq. (16).
%F (1/8)* this = 2 -(15/16) * 2F1(3/4,5/4 ; 3 ;1). [Bailey eq (4.5)]
%F Integral_(x= 0.. 2*Pi) sqrt(1+sin x) dx = 5.6569... = 2*Pi*(1-this /16). [Evans N_1]
%F Equals 16 - 32*sqrt(2)/Pi. - _Amiram Eldar_, Sep 06 2024
%e 1.59493894148...
%p 16-15/2*hypergeom([3/4,5/4],[3],1) ; evalf(%) ;
%t RealDigits[16 - 32*Sqrt[2]/Pi, 10, 120][[1]] (* _Amiram Eldar_, Sep 06 2024 *)
%K cons,nonn
%O 1,2
%A _R. J. Mathar_, Sep 05 2024