Fourier series and even/odd functions

  1. [SOLVED] Fourier series and even/odd functions

    1. The problem statement, all variables and given/known data
    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?
     
    Last edited: Mar 29, 2008
  2. jcsd
  3. Dick

    Dick 25,887
    Science Advisor
    Homework Helper

    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.
     
  4. Ahh, I see.

    You have helped me very much lately. Thank you.
     
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