1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    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!

Fourier transform of the hyperbolic secant function

  1. Jan 6, 2013 #1
    1. The problem statement, all variables and given/known data

    Hi there!!

    I'm just trying to figure out the Fourier transform of the hyperbolic secant function... I already know the outcome:

    4[itex]\sum\ ((-1)^n*(1+2n))/(ω^2*(2n+1)^2) [/itex]

    But sadly, I cannot figure out how to work round to it! :( maybe one of you could help me...

    2. Relevant equations

    I was thinking of using the geometric series 1/(1+q) = [itex]\sum (-q)^n [/itex] for q = e^(-2*x), as the hyperbolic secant is 2*e^(-x)/(1+e^(-2*x)) .
    And then you need to multiply the hyperbolic secant with e^(iωt) and integrate from -∞
    up to ∞.
    But... frankly, that was it. I have never managed to come to the result
    4[itex]\sum\ ((-1)^n*(1+2n))/(ω^2*(2n+1)^2) [/itex]

    Maybe someone could give me a hint to to solve this problem...? :)

    Lots of greetings,
    Last edited: Jan 6, 2013
  2. jcsd
  3. Jan 6, 2013 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    f(x) = sech(x) is an even function, so the FT of f is (g + conjugate g), where ##g = \int_0^{\infty} f(x) e^{i \omega x} \, dx.## Apply this term-by-term to
    [tex] \text{sech}(x) = 2 \sum (-1)^n e^{-(2n+1)x}.[/tex]
  4. Jan 6, 2013 #3

    Aaahh, wonderful!! Thank you!

    Just one more question... where does the Term [tex](-1)^n[/tex] come from?

    Thanks a lot!!
  5. Jan 6, 2013 #4


    User Avatar
    Science Advisor
    Homework Helper

    It comes from the expansion of 1/(1+a)=1-a+a^2-a^3+... It's the factor that alternates sign.
    Last edited: Jan 6, 2013
  6. Jan 6, 2013 #5
    Oh yes, sure!!! Now I got it :)

    Thank you very much!! :) And have a nice evening!!

  7. Jan 6, 2013 #6


    User Avatar
    Science Advisor
    Homework Helper

    You're welcome. Nice evening to you too. Note I goofed on the post. I meant 1/(1+a) not 1/(1-a).
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Fourier transform hyperbolic Date
Fourier transform of exponential function Thursday at 9:39 AM
Diffusion equation in polar coordinates Wednesday at 2:48 PM
Fourier transform between variables of different domains Dec 19, 2017
Fourier Transform Dec 11, 2017
Fourier transform (FT) of a Hyperbolic secant Sep 18, 2008