OFFSET
1,2
LINKS
W. N. Bailey, Contiguous hypergeometric functions of the type 3F2, Proc. Glasg. Math. Ass. 2 (1954) 62-65.
James D. Evans, Evaluation of four irrational definite sine integrals using residue theory, Appl. Math. Comp 36 (1990) 163-172, eq. (16).
FORMULA
(1/8)* this = 2 -(15/16) * 2F1(3/4,5/4 ; 3 ;1). [Bailey eq (4.5)]
Integral_(x= 0.. 2*Pi) sqrt(1+sin x) dx = 5.6569... = 2*Pi*(1-this /16). [Evans N_1]
Equals 16 - 32*sqrt(2)/Pi. - Amiram Eldar, Sep 06 2024
EXAMPLE
1.59493894148...
MAPLE
16-15/2*hypergeom([3/4, 5/4], [3], 1) ; evalf(%) ;
MATHEMATICA
RealDigits[16 - 32*Sqrt[2]/Pi, 10, 120][[1]] (* Amiram Eldar, Sep 06 2024 *)
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
R. J. Mathar, Sep 05 2024
STATUS
approved