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Fourier series and even/odd functions

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Mar29-08, 11:03 AM
P: 1,863
1. The problem statement, all variables and given/known data
I found the solution to a PDE in this thread: (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?
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Mar29-08, 11:30 AM
Sci Advisor
HW Helper
P: 25,251
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.
Mar29-08, 11:31 AM
P: 1,863
Ahh, I see.

You have helped me very much lately. Thank you.

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