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Fourier transform of signum function*exponential

1. The problem statement, all variables and given/known data

Using properties of the Fourier transform, calculate the Fourier transform of: sgn(x)*e^(-a*abs(x-2))

2. Relevant equations

FT(f(x))= integral from -∞ to +∞ of f(x)*e^(-iwx) dx

3. The attempt at a solution

I've realised that with the signum function, the boundaries of the integral can be reduced to: integral from 0 to 4 of e^(-a*abs(x-2))*e^(-iwx) dx from 0 to 4. However I'm guessing that I just can't use the properties and theorems related to fourier transforms as the integral does not have the same boundaries as the original... Maybe I just souldn't have changed the integral...Any help?

Sorry for not using latex I really don't understand it =/
 

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vela

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What was your reasoning that you could change the limits to 0 and 4?
 
The function from -∞ to 0 is the opposite of the function from 4 to ∞ so they cancel out. Can I do that?
 

vela

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That's true for f(x) but not for f(x)e-iωx. The exponential messes things up.
 
Ahh indeed :(... How should I approach it then?
 

vela

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I suggest using the convolution theorem. Or just grind it out — it doesn't look like there's anything tricky going on.
 

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