Why Is the Fourier Sine Series Not Defined at Discontinuous Points?

[zorro]

Why is Fourier sine series of any function satisfying Dirichlet's theorem, not defined on the discontinuous points whereas we define it for Fourier cosine series?

ex - sine series of f(x) = cosx, 0<=x<=∏ is defined on 0<x<∏

whereas cosine series of f(x) = sinx, 0<=x<=∏ is defined on 0<=x<=∏

 

[LCKurtz]

Why is Fourier sine series of any function satisfying Dirichlet's theorem, not defined on the discontinuous points whereas we define it for Fourier cosine series?

ex - sine series of f(x) = cosx, 0<=x<=∏ is defined on 0<x<∏

whereas cosine series of f(x) = sinx, 0<=x<=∏ is defined on 0<=x<=∏

Not sure what you are getting at. The half range Fourier series you mention both converge for all x. In the first case the FS converges to the average of the right and left hand limits at x = 0 of the odd extension of cos(x). In the second case the FS converges to sin(0) = 0 at x = 0. That is because the even extension of sin(x) is continuous at x=0 while the odd extension of cos(x) is not continuous at x = 0.