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%I #30 Mar 10 2023 09:08:37
%S 1,8,5,1,9,3,7,0,5,1,9,8,2,4,6,6,1,7,0,3,6,1,0,5,3,3,7,0,1,5,7,9,9,1,
%T 3,6,3,3,4,5,8,0,9,7,2,8,9,8,1,1,5,4,9,0,9,8,0,4,7,8,3,7,8,1,8,7,6,9,
%U 8,1,8,9,0,1,6,6,3,4,8,3,5,8,5,3,2,7,1,0,3,3,6,5,0,2,9,5,4,7,5,7,7,0,1,6,8
%N Decimal expansion of Integral_{x=0..Pi} (sin(x)/x) dx.
%C Also known as Gibbs constant.
%C Integral(sin(x)/x dx) = x - x^3/(3*3!) + x^5/(5*5!) - x^7/(7*7!) + ... - _Harry J. Smith_, May 01 2009
%C Also called Wilbraham-Gibbs (or Gibbs-Wilbraham) constant after the English mathematician Henry Wilbraham (1825-1883) and the American scientist Josiah Willard Gibbs (1839-1903). - _Amiram Eldar_, Jun 15 2021
%C This constant and its negative are respectively the maximum and the minimum value of Integral_{0..x} (sin(t)/t) dt for real x. - _Jianing Song_, Mar 10 2023
%D Steven Finch, Mathematical Constants, Cambridge University Press, 2003, section 4.1, "Gibbs-Wilbraham Constant", pp. 248-250.
%H Harry J. Smith, <a href="/A036792/b036792.txt">Table of n, a(n) for n = 1..20000</a>
%H J. Willard Gibbs, <a href="https://doi.org/10.1038/059606a0">Fourier's Series</a>, Nature, Vol. 59, No. 1539 (1899), p. 606.
%H Simon Plouffe, <a href="https://web.archive.org/web/20050310182110/http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap44.html">Gibbs, Si(Pi) or the Gibbs Constant to 1024 places</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Wilbraham-GibbsConstant.html">Wilbraham-Gibbs Constant</a>.
%H Henry Wilbraham, <a href="https://books.google.com/books?id=JrQ4AAAAMAAJ&pg=PA198">On a certain periodic function</a>, The Cambridge and Dublin Mathematical Journal, Vol. 3 (1848), pp. 198-201.
%e 1.85193705198246617036105337015799136334580972898115...
%t RealDigits[ N[ SinIntegral[Pi], 110]] [[1]]
%o (PARI) { default(realprecision, 20080); y=0; x=Pi; m=x; x2=x*x; n=1; nf=1; s=1; while (x!=y, y=x; n++; nf*=n; n++; nf*=n; m*=x2; s=-s; x+=s*m/(n*nf)); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b036792.txt", n, " ", d)); } \\ _Harry J. Smith_, May 01 2009
%Y Cf. A036790.
%K nonn,cons
%O 1,2
%A _N. J. A. Sloane_
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