1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Justify an equality involving hyperbolic cosine and Fourier series

  1. Sep 25, 2013 #1
    1. The problem statement, all variables and given/known data
    The problem:

    Justify the following equalities:
    [tex]\cot x = i\coth (ix) = i \sum^\infty_{n=-\infty} \frac{ix}{(ix)^2+(n\pi)^2}=\sum^\infty_{n=-\infty}\frac{x}{x^2+(n\pi)^2}[/tex]

    I am trying to figure out how to start this. When I insert the Euler identity of [itex]
    \coth[/itex] (using the formula for complex Fourier series) I end up with: [tex]c_n = \frac{1}{2\pi}\int^{\pi}_{-\pi}\frac{e^{x} + e^{-x}}{e^{x} - e^{-x}}e^{inx} \ dx = \frac{1}{2\pi} \int^{\pi}_{-\pi}\frac{e^{(1+in)x} + e^{-(1+in)x}}{e^{x} - e^{-x}} \ dx[/tex]

    which is one ugly integral. So my question is a) did I make a mistake in the starting point and b) can this integral be simplified in some way that's better? Or is there some stupidly silly pattern I should be recognizing here? (I considered treating the integral as [itex]\frac{1}{2\pi} \int^{\pi}_{-\pi} \frac{u}{du}[/itex] or something like it).

    I suspect I am missing something obvious.

    Thanks for any help. This one I think involves doing out a complex Fourier series for cosh or sinh, which might be simpler. But I am not sure.
     
  2. jcsd
  3. Sep 25, 2013 #2
    Here's a thought. Are you familiar with breaking up a fraction like ##\frac{A + B} {C + D}##? Just say ## \frac {x}{C} + \frac {y}{D}## = ##\frac{A + B} {C + D}##. On the left side, add up the two fractions as you did in elementary school and solve for x and y.

    If you do that to your integrand, you will wind up with the sum of two exponentials, and I think you can just integrate it. Whether that trick gets you to the right series I don't know, but it's worth a shot.

    I'm also a little worried about your summations -- when you multiply through by i don't you get a -x in the numerator?

    Finally -- are you sure this is a Fourier series problem? Is it in that chapter of some book? I am not above expanding cotx in a Taylor's series and seeing if that can be kicked around to get what you need.
     
  4. Sep 26, 2013 #3
    It's in the Fourier Series chapter, and the section on complex Fouriers, so presumably they want something like that. There's an earlier problem in the group where they tell you to get a complex Fourier of cosh, and that would be used here, but I am not entirely sure how that would work.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Justify an equality involving hyperbolic cosine and Fourier series
  1. Fourier Cosine Series (Replies: 7)

  2. Fourier Cosine Series (Replies: 6)

Loading...