Formal Mathematical Description of The Phenomenon
Let be a piecewise continuously differentiable function which is periodic with some period . Suppose that at some point, the left limit and right limit of the function differ by a non-zero gap :
For each positive integer N ≥ 1, let SN f be the Nth partial Fourier series
where the Fourier coefficients are given by the usual formulae
Then we have
and
but
More generally, if is any sequence of real numbers which converges to as, and if the gap a is positive then
and
If instead the gap a is negative, one needs to interchange limit superior with limit inferior, and also interchange the ≤ and ≥ signs, in the above two inequalities.
Read more about this topic: Gibbs Phenomenon
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