Is there a solution to this integral? -exp(-a*abs(x))*exp(i*(k0-k)*x)*dx

[quasar_4]

Hi everyone. Maple and I have collectively racked our brains and I've tried most of the integration techniques I know. Does anyone know the solution to the integral

$$\int$$ exp(-a*abs(x))*exp(i*(k0-k)*x)*dx

from -infinity to infinity (not sure how to get the limits over the integral sign here in this text box)?

There might be a good change of variables, but my brain is now too fried to think of it. Or does this baby just not have a nice solution? anyone?

 

[esbo]

Not me :O)

 

[uart]

What? That's trivial if you just split the integral into two parts, one part from -infinity to zero and the other from 0 to infinity as it let's you get rid of the annoying abs(x).

I got $$\frac{1}{a-bi} + \frac{1}{a+bi} = \frac{2a}{a^2+b^2}$$

BTW. $$b = k_0 - k$$ in my solution.

 

[quasar_4]

Haha, yes, I realized that once I got home and felt REALLY dumb for posting the previous msg. I think in fact you can use trig identities and Euler's relation as well to split into a sin and cos part, then the sin drops out (since it's over a symmetric interval) and you can take twice the integral of the cos part from 0-infinity. That hadn't worked at the time, but it turns out I was being brain-dead and forgetting to drop my abs. Duh! That's what I get for doing homework on 2 hours of sleep...