Fourier series and even/odd functions

[Niles]

[SOLVED] Fourier series and even/odd functions

Homework Statement

I found the solution to a PDE in this thread: https://www.physicsforums.com/showthread.php?t=224902 (not important)

The solution is the sum of u(rho, phi) = [A_n*cos(n*phi)+B_n*sin(n*phi)]*rho^n.

I have to find the general solution, where rho=c, so I equal rho = c, and I am told that u in this point equals sin(phi/2) when phi is between 0 and 2*pi.

I must find the Fourier-coefficients (since it is a Fourier-series).

My questions are:

Since sin(phi/2) is an ODD function, can I discard A_n and just find B_n? That is what I would do, but in the solutions in the back of my book they find A_n as well. Why is that?! The book even says that for an odd function, the Fourier-series only contains sine, so A_n can be discarded, but they still find it. Can you explain to me why A_n must be found as well?

 

[Dick]

The integral of an even function times and odd function will generally vanish only if you are integrating over an interval symmetric around the origin, like [-L,L]. Your interval here is [0,2pi]. The A_n's don't automatically vanish.

 

[Niles]

Ahh, I see.

You have helped me very much lately. Thank you.