The Square Wave Example
We now illustrate the above Gibbs phenomenon in the case of the square wave described earlier. In this case the period L is, the discontinuity is at zero, and the jump a is equal to . For simplicity let us just deal with the case when N is even (the case of odd N is very similar). Then we have
Substituting, we obtain
as claimed above. Next, we compute
If we introduce the normalized sinc function, we can rewrite this as
But the expression in square brackets is a numerical integration approximation to the integral (more precisely, it is a midpoint rule approximation with spacing ). Since the sinc function is continuous, this approximation converges to the actual integral as . Thus we have
which was what was claimed in the previous section. A similar computation shows
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