Let a piecewise smooth function with only finitely many discontinuities (which are all jumps) be defined on with Fourier series
(1)
| |||
(2)
|
(3)
|
Let a discontinuity be at , with
(4)
|
so
(5)
|
Define
(6)
|
and let be the first local minimum and the first local maximum of on either side of . Then
(7)
|
(8)
|
where
(9)
| |||
(10)
| |||
(11)
|
(OEIS A036792). Here, is the sinc function and is the sine integral.
The Fourier series of therefore does not converge to and at the ends, but to and . This phenomenon was observed by Wilbraham in 1848 and Gibbs in 1899. Although Wilbraham was the first to note the phenomenon, the constant is frequently (and unfairly) credited to Gibbs and known as the Gibbs constant.
A related constant sometimes also called the Gibbs constant is
(12)
|
(OEIS A036793; Le Lionnais 1983).