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Integral of exponential involving sines and cosines

  1. Mar 24, 2007 #1

    Can someone help me solve the following integral

    \int_0^{2\pi} exp[-i(G_x \cos\theta + G_y \sin\theta] d\theta

    I've tried splitting the exponential into a product of two exponetials and rewriting the exponentials in terms of sines and cosines. But I always end up getting stuck. Some of my rewritings ended up looking close to a Bessel function but it's just not quite the same. Can someone just give me a hint on where to start?

    Im not sure about the limits, they might have to be from [tex]-\pi[/tex] to [tex]\pi[/tex] but I don't believe that should change anything.

    Thanks in advance
  2. jcsd
  3. Mar 24, 2007 #2
    Probably the answer will look like some combination of Bessel Js.

    J_{n}(x) = \int_{0}^{2\pi}\frac{d\phi}{2\pi}e^{-ix\sin\phi + in\phi}

    There is an expansion:

    e^{-ix\sin\phi} = \sum_{k=-\infty}^{\infty}J_{k}(x)e^{-ik\phi}

    So in principle you can expand each exponentional term into Fourier harmonics and evaluate the angular integral, then resum the resulting expression. There is probably an easier way though....
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