Let a piecewise smooth function 
with only finitely many discontinuities (which are all jumps)
be defined on ![[-pi,pi]](/images/equations/Wilbraham-GibbsConstant/Inline2.svg)
with Fourier series
 |  |  |
(1)
|
 |  |  |
(2)
|
![S_n(f,x)=1/2a_0+{sum_(k=1)^n[a_kcos(kx)+b_ksin(kx)]}.](/images/equations/Wilbraham-GibbsConstant/NumberedEquation1.svg) |
(3)
|
Let a discontinuity be at 
, with
 |
(4)
|
so
![D=[lim_(x->c^-)f(x)]-[lim_(x->c^+)f(x)]>0.](/images/equations/Wilbraham-GibbsConstant/NumberedEquation3.svg) |
(5)
|
Define
![phi(c)=1/2[lim_(x->c^-)f(x)+lim_(x->c^+)f(x)],](/images/equations/Wilbraham-GibbsConstant/NumberedEquation4.svg) |
(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).
