Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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....
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?