Fourier transform of signum function*exponential

[quebecois22]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

I've realized 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 =/

 

[vela]

What was your reasoning that you could change the limits to 0 and 4?

 

[quebecois22]

The function from -∞ to 0 is the opposite of the function from 4 to ∞ so they cancel out. Can I do that?

 

[vela]

That's true for f(x) but not for f(x)e^-iωx. The exponential messes things up.

 

[quebecois22]

Ahh indeed :(... How should I approach it then?

 

[vela]

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