A036792 Decimal expansion of Integral_{x=0..Pi} (sin(x)/x) dx.

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, 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, 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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• cons

Comments

Also known as Gibbs constant.

Integral(sin(x)/x dx) = x - x^3/(3*3!) + x^5/(5*5!) - x^7/(7*7!) + ... - Harry J. Smith, May 01 2009

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

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

Examples

			1.85193705198246617036105337015799136334580972898115...
		

References

• Steven Finch, Mathematical Constants, Cambridge University Press, 2003, section 4.1, "Gibbs-Wilbraham Constant", pp. 248-250.
• Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 38, page 366.

Links

• Harry J. Smith, Table of n, a(n) for n = 1..20000
• J. Willard Gibbs, Fourier's Series, Nature, Vol. 59, No. 1539 (1899), p. 606.
• Simon Plouffe, Gibbs, Si(Pi) or the Gibbs Constant to 1024 places.
• Eric Weisstein's World of Mathematics, Wilbraham-Gibbs Constant.
• Henry Wilbraham, On a certain periodic function, The Cambridge and Dublin Mathematical Journal, Vol. 3 (1848), pp. 198-201.

Crossrefs

Cf. A036790.

Programs

Mathematica

RealDigits[ N[ SinIntegral[Pi], 110]] [[1]]

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