# Homework Help: Fourier transform of the hyperbolic secant function

1. Jan 6, 2013

### Marie_Curie

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$\sum\ ((-1)^n*(1+2n))/(ω^2*(2n+1)^2)$

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) = $\sum (-q)^n$ 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$\sum\ ((-1)^n*(1+2n))/(ω^2*(2n+1)^2)$

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

Lots of greetings,
Marie

Last edited: Jan 6, 2013
2. Jan 6, 2013

### Ray Vickson

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
$$\text{sech}(x) = 2 \sum (-1)^n e^{-(2n+1)x}.$$

3. Jan 6, 2013

### Marie_Curie

Aaahh, wonderful!! Thank you!

Just one more question... where does the Term $$(-1)^n$$ come from?

Thanks a lot!!
MARIE

4. Jan 6, 2013

### Dick

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
5. Jan 6, 2013

### Marie_Curie

Oh yes, sure!!! Now I got it :)

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

_MARIE_

6. Jan 6, 2013

### Dick

You're welcome. Nice evening to you too. Note I goofed on the post. I meant 1/(1+a) not 1/(1-a).