Solution of integral involving exponential and absolute values

Question

I want to solve this integral $\int_0 ^\infty e^{-iwx}e^{-α|x|} dx$

Any ideas on how to solve it?

Answer 1

$$\int_0 ^\infty e^{-iwx}e^{-\alpha|x|} dx = \int_0 ^\infty e^{-(iw +\alpha) x } dx$$

Assuming that $\Re(\alpha - iw)>0$ we write this integral as

$$\int_0 ^\infty e^{-(iw +\alpha) x } dx = \int_0 ^\infty \frac{d (e^{-(iw +\alpha) x })}{dx} \frac{dx}{-(iw +\alpha) } = \frac{1}{ iw +\alpha}.$$

Answer 2

$x \ge 0$ so we have that the integral is equal to

$$\int_0^{\infty} dx \, e^{-i w x} e^{- \alpha x} = \frac1{\alpha + i w} $$

(assuming $\alpha \gt 0$).