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Wilbraham-Gibbs Constant


Let a piecewise smooth function f with only finitely many discontinuities (which are all jumps) be defined on [-pi,pi] with Fourier series

a_k=1/piint_(-pi)^pif(t)cos(kt)dt
(1)
b_k=1/piint_(-pi)^pif(t)sin(kt)dt
(2)
 S_n(f,x)=1/2a_0+{sum_(k=1)^n[a_kcos(kx)+b_ksin(kx)]}.
(3)

Let a discontinuity be at x=c, with

 lim_(x->c^-)f(x)>lim_(x->c^+)f(x),
(4)

so

 D=[lim_(x->c^-)f(x)]-[lim_(x->c^+)f(x)]>0.
(5)

Define

 phi(c)=1/2[lim_(x->c^-)f(x)+lim_(x->c^+)f(x)],
(6)

and let x=x_n<c be the first local minimum and x=xi_n>c the first local maximum of S_n(f,x) on either side of x_n. Then

 lim_(n->infty)S_n(f,x_n)=phi(c)+D/piG^'
(7)
 lim_(n->infty)S_n(f,xi_n)=phi(c)-D/piG^',
(8)

where

G^'=int_0^pisincthetadtheta
(9)
=Si(pi)
(10)
=1.851937052...
(11)

(OEIS A036792). Here, sinc(x)=sinx/x is the sinc function and Si(x) is the sine integral.

The Fourier series of y=x therefore does not converge to -pi and pi at the ends, but to -2G^' and 2G^'. This phenomenon was observed by Wilbraham in 1848 and Gibbs in 1899. Although Wilbraham was the first to note the phenomenon, the constant G^' is frequently (and unfairly) credited to Gibbs and known as the Gibbs constant.

A related constant sometimes also called the Gibbs constant is

 G=2/piG^'=1.17897974447216727...
(12)

(OEIS A036793; Le Lionnais 1983).


See also

Apodization, Fourier Series, Gibbs Phenomenon

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References

Carslaw, H. S. Introduction to the Theory of Fourier's Series and Integrals, 3rd ed. New York: Dover, 1930.Finch, S. R. "Gibbs-Wilbraham Constant." §4.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 248-250, 2003.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 36 and 43, 1983.Sloane, N. J. A. Sequences A036792 and A036793 in "The On-Line Encyclopedia of Integer Sequences."Zygmund, A. G. Trigonometric Series 1, 2nd ed. Cambridge, England: Cambridge University Press, 1959.

Referenced on Wolfram|Alpha

Wilbraham-Gibbs Constant

Cite this as:

Weisstein, Eric W. "Wilbraham-Gibbs Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Wilbraham-GibbsConstant.html

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