Integrating Fourier Transform of Exponential Function

[Swatch]

I am trying to solve this Fourier problem where I have to integrate
∫f(x) * exp(-i§x) dx from -∞ to ∞ , where f(x) = exp(-sgn(x))
I tried breaking the function into two pieces where x is from -∞ to 0 and from 0 to ∞ where f(x) would then be exp(x) and exp(-x) and integrating two functions, but that didn't seem to be working. The the other way I can think of is trying to integrate
exp(-i§x) * exp(-sgn(x)), but I'm not sure if that is possible. Could anyone please give me a hint?

 

[benorin]

What is § ? a constant, an operator, or a function?

 

[Swatch]

it is (n*PI)/L ,where 2L is one period which I just treat as a constant

 

[HallsofIvy]

L is one period of what? If your function, f, is periodic, then you need only integrate over one period. In that case, you are talking about a Fourier Series, not a Fourier transform and probably would find it easier to use $sin(\frac{n\pi}{L}t)$ and [tex]cos(\frac{n\pi}{L}t)[/itex] rather than complex exponentials.

If is not periodic, then you need the Fourier transform $\int_{-infty}^\infty f(x)e^{-ixt}dt$.