Wilbraham-Gibbs Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

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

$$a_k = {1\over\pi}\int_{-\pi}^\pi f(t)\cos(kt)\,dt$$ (1)

$$b_k = {1\over\pi}\int_{-\pi}^\pi f(t)\sin(kt)\,dt,$$ (2)

$$S_n(f,x)={\textstyle{1\over 2}}a_0+\left\{{\sum_{k=1}^n [a_k\cos(kx)+b_k\sin(kx)]}\right\}.$$ (3)

Let a discontinuity be at $x=c$, with

$$\lim_{x\to c^-} f(x)>\lim_{x\to c^+} f(x),$$ (4)

so

$$D\equiv \left[{\,\lim_{x\to c^-} f(x)}\right]-\left[{\,\lim_{x\to c^+} f(x)}\right]>0.$$ (5)

Define

$$\phi(c)={1\over 2} \left[{\,\lim_{x\to c^-} f(x)+\lim_{x\to c^+} f(x)}\right],$$ (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

$$\lim_{n\to\infty} S_n(f,x_n)=\phi(c)+{D\over\pi}G'$$ (7)

$$\lim_{n\to\infty} S_n(f,\xi_n)=\phi(c)-{D\over\pi}G',$$ (8)

where

$$G'\equiv \int_0^\pi \mathop{\rm sinc}\nolimits \theta\,d\theta=1.851937052\ldots.$$ (9)

Here, $\mathop{\rm sinc}\nolimits x\equiv \sin x/x$ is the Sinc Function. 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 (1848) and Gibbs (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

$$G\equiv {2\over\pi} G'={2\over\pi}\int_0^\pi \mathop{\rm sinc}\nolimits x\,dx =1.17897974447216727\ldots$$ (10)

(Le Lionnais 1983).

References

Carslaw, H. S. Introduction to the Theory of Fourier's Series and Integrals, 3rd ed. New York: Dover, 1930.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/gibbs/gibbs.html

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 36 and 43, 1983.

Zygmund, A. G. Trigonometric Series 1, 2nd ed. Cambridge, England: Cambridge University Press, 1959.