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Complex Fourier Series Problem

  1. Apr 14, 2008 #1
    1. The problem statement, all variables and given/known data

    Having shown that:

    [tex]f(x) = e^{ax} = \frac{sinh(a\pi)}{\pi}\sum^{\infty}_{n=-\infty}\frac{(-1)^{n}(a+in)}{(a^{2}+n^{2})}e^{inx}[/tex]

    on [tex](-\pi,\pi)[/tex] with 'a' a real constant

    Deduce that:

    [tex]\pi coth(a\pi) = \frac{1}{a} + \sum^{\infty}_{n=1}\frac{2a}{(a^{2}+n^{2})} [/tex]

    2. Relevant equations

    [tex]sinhx = \frac{e^{ax} - e^{-ax}}{2}[/tex]

    [tex]coshx = \frac{e^{ax} + e^{-ax}}{2}[/tex]

    [tex]cothx = \frac{coshx}{sinhx}[/tex]

    3. The attempt at a solution

    I was able to derive the very top equation; I then set [tex]x=\pi[/tex] (in the top equation) so that it reduced to:

    [tex]e^{a\pi} = \frac{sinh(a\pi)}{\pi}\sum^{\infty}_{n=-\infty}\frac{(a+in)}{(a^{2}+n^{2})}e^{inx}[/tex]

    I know that by fiddling around with the definition of cothx I can get fairly close to what I want, but my major issue here is the change of the 'interval' of summation between what I have shown and what I am aiming to show. Any help would be greatly appreciated.
    Last edited: Apr 14, 2008
  2. jcsd
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