The Wilbraham-Gibbs constant is the constant governing the limiting overshoot of Fourier partial sums at a jump
discontinuity. Let a real-valued piecewise smooth function 
be defined on 
and extended periodically, with only finitely many
discontinuities in each period, all finite jumps. Its Fourier
series has coefficients
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(1)
|
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(2)
|
For a nonnegative integer 
, the 
th partial sum is
![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 maximum of 
to the left of 
and 
the first local minimum
to the right of 
(Finch 2003, p. 248). Then
 |
(7)
|
 |
(8)
|
where
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(9)
|
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(10)
|
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(11)
|
(OEIS A036792). Here, 
for 
, with 
, is the sinc function
and 
is the sine integral.
For the periodic extension of 
from 
, every Fourier partial sum
equals 0 at 
.
The adjacent extrema inside this interval tend to 
near 
and 
near 
, rather than to the one-sided limits 
and 
. These are moving-extremum limits, not the values of the
series at the endpoints. Finch (2003, pp. 249-250) documents the observations
of Wilbraham in 1848 and Gibbs in 1898 and 1899. The constant 
is also known as the Gibbs constant.
A related constant sometimes also called the Gibbs constant is
 |
(12)
|
(OEIS A036793; Le Lionnais 1983).
