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Decimal expansion of Pi/2*(Pi^2/12 + (log(2))^2).
3

%I #27 Aug 23 2024 05:07:29

%S 2,0,4,6,6,2,2,0,2,4,4,7,2,7,4,0,6,4,6,1,6,9,6,4,1,0,0,8,1,7,6,9,7,3,

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

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

%N Decimal expansion of Pi/2*(Pi^2/12 + (log(2))^2).

%C The value of the integral_{x=0..Pi/2} log(sin(x))^2 dx. The value of sqrt(Pi)/2*(d^2/da^2(gamma((a+1)/2)/gamma(a/2+1))) at a=0. - _Seiichi Kirikami_ and _Peter J. C. Moses_, Oct 07 2011

%D I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 3.621.1

%H G. C. Greubel, <a href="/A196877/b196877.txt">Table of n, a(n) for n = 1..10000</a>

%H K. S. Kolbig, <a href="https://doi.org/10.1090/S0025-5718-1983-0689472-3">On the integral int_0^Pi/2 log^n cos x log^p sin x dx</a>, Math. Comp. 40 (162) (1983) 565-570, r_{2,0}

%F Equals A019669*(A072691 + A002162^2).

%F Equals Integral_{x=0..1} log(x)^2/sqrt(1-x^2) dx. - _Amiram Eldar_, May 27 2023

%e 2.04662202447274064616964100817...

%t RealDigits[N[Pi/2 (Pi^2/12 + Log[2]^2),150] [[1]]

%o (PARI) Pi/2*(Pi^2/12+(log(2))^2) \\ _Michel Marcus_, Jan 13 2015

%Y Cf. A002162, A019669, A072691,A173623, A196878.

%K cons,nonn

%O 1,1

%A _Seiichi Kirikami_, Oct 07 2011