I've seen the Fourier Sine Transform defined in different ways.
Specifically:
$$F_s(\alpha) = \int_{-\infty}^{\infty}f(x)\sin(\alpha x)\space dx,$$
and:
$$F_s(\alpha) = \int_{0}^{\infty}f(x)\sin(\alpha x)\space dx$$
The difference is obviously that in the second case the lower limit is zero rather than negative infinity.
I don't know which one to use in my specific case: $f(x) = e^{-mx}$, $m >0$. Is there some constraint on the function in the second case that allows us to then increase the lower limit to zero?
HINT:
For $m>0$ and $\alpha \ne0$, does the integral $\int_{-\infty }^0 \sin(\alpha x)e^{-mx}\,dx$ converge?
And for $x<0$, is $f(x)=0$?