Fourier transform $e^{-bx}\theta(x-a)$?

Question

I'm trying to find the fourier transform of $e^{-bx}\theta(x-a)$ where $\theta$ is the step function ($\theta(x)=1$ if $x>0$, $\theta(x)=0$ if $x<0$, $\theta(x)=.5$ if $x=0$), but I can't get past how to integrate with $\theta$. I tried using the definition of $\theta$ as the integral of the delta function, but I didn't get anywhere. Does anyone know how to approach this?

Answer 1

The FT is

$$F(k) = \int_a^{\infty} dx \, e^{-b x} \, e^{i k x} = \frac1{b-i k} e^{-(b - i k) a}$$