Fourier Series at Discontinuities

[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
Hermes10

 

[mathman]

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