# Fourier Series at Discontinuities

1. Dec 10, 2011

### Hermes10

Dear all,

I am wondering why the Fourier Series converges at a finite discontinuity of a periodic function at 1/2*(y1+y2) at the point f(x1), where x1 is the point at which the discontinuity occurs and y1 is the limiting value for the function when we approach x=x1 from one side and y2 is the limiting value when we approach x=x1 from the other side?

Say, in a particular case y2 is 5 and y1 is 2, shouldn't the Fourier series converge to 1/2*(5-2)? I would have though that the Fourier series just converges at the midpoint between y1 and y2 on the graph that is if you draw the function I would have draw the value for x1 at which the discontinuity occurs to be in the middle of the two limiting values. Is that correct?

All the Best,
Hermes10

2. Dec 10, 2011

### mathman

Your idea is correct, except for an error (typo?) y2=5 and y1=2 gives 1/2(5+2) as the midpoint.