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Math Help - [SOLVED] fourier series of exponential function

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    [SOLVED] fourier series of exponential function

    Edit: sorry, figured it out: I should have been expecting the coefficients to have an imaginary part

    Hi, I am doing some revision and am trying to find the fourier series for the exponential function between -\pi and \pi. I am fairly sure all the coefficients should be real but I keep getting stuck with an imaginary part.

    Please tell me where I am wrong:

    e^t = \sum_{n=-\infty}^{\infty}f_ne^{int}<br />
where
    f_n = \frac{1}{2\pi}\int_{-\pi}^{\pi}e^te^{-int}dt<br />
    f_n = \frac{1}{2\pi}\int_{-\pi}^{\pi}e^{t-int}dt
    f_n = \frac{1}{2\pi}\int_{-\pi}^{\pi}e^{t(1-in)}dt
    f_n = \frac{1}{2\pi}\frac{1}{1-in}[e^te^{-int}]_{-\pi}^{\pi}
    f_n = \frac{1}{2\pi}\frac{1}{1-in}[e^{\pi}e^{-i\pi n}-e^{-\pi}e^{i\pi n}]
    f_n = \frac{1}{2\pi}\frac{1}{1-in}[e^{\pi}(-1)^n-e^{-\pi}(-1)^n]
    f_n = \frac{1}{2\pi}\frac{1}{1-in}(-1)^n2\sinh(\pi)
    f_n = \frac{1}{\pi}\frac{1}{1+n^2}(1+in)(-1)^n\sinh(\pi)

    Thanks a lot
    Last edited by badgerigar; October 12th 2008 at 11:25 PM.
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