$\int\limits_{\mathbb{R}} e^{-|x|}e^{-ix\xi}dx$

Question

I can't compute this $\int\limits_{\mathbb{R}} e^{-|x|}e^{-ix\xi}dx$. I have separate it into 2 integrals but i can't continue.

Answer 1

$$\int_{\mathbb{R}}e^{|x|}e^{-ix\xi} \; dx=\int_{0}^{\infty}e^{-x}e^{-ix\xi}dx+\int_{-\infty}^{0}e^{-x}e^{-ix\xi}dx$$

First integral:

$$\int_{0}^{\infty}e^{x}e^{-ix\xi}dx=\int_{0}^{\infty}e^{x(-1-i\xi)}dx=\lim_{a \to \infty}\int_{0}^{a}e^{x(-1-i\xi)}dx=\left.\lim_{a \to \infty}\frac{e^{x(-1-i\xi)}}{(-1-i\xi)} \right|_{0}^{a}=\lim_{a \to \infty}\frac{e^{a(-1-i\xi)}}{(-1-i\xi)} - \frac{e^{0}}{(-1-i\xi)}=-\frac{e^{0}}{(-1-i\xi)}$$

Can you do similar computation for the second integral?