Writing Fourier Series for Open and Closed Intervals

[aaaa202]

In a rigorous mathematical course I am talking, it seems to make a difference when I am given a function f and need to write its Fourier series, whether it is defined on [0,2∏] or [0,2∏). What difference does it make for my series whether it is an open or a closed interval?

 

[micromass]

It doesn't make a difference. If two function differ in one point only (or if one function is not defined in one point), then the Fourier series will still be the same thing. The reason for this is that Fourier series only depend on integrals such as

\int_0^\pi f(x)\sin(nx)dx

But if $f$ changes in one point, then the integral will of course remain the same thing.