How to find Fourier coefficients of $f(x)=e^{-|x|}$
By definition $\hat f(\xi)=\int_{\mathbb{R}}e^{-(|x|+i\xi x)} \ dx$
let $\alpha<0, \beta>0\ $and let$$I(\alpha,\beta)=\int_{\alpha}^{\beta} e^{-(|x|+i\xi x)} \ dx$$ Then $$I=\frac{2}{1+\xi^2}-\frac{e^{(1-i\xi)\alpha}}{1-i\xi}-\frac{e^{-(1+i\xi)\beta}}{1+i\xi}$$
Now taking $\alpha\to -\infty$ and $\beta\to \infty$ we get last two terms goes to zero as their modulus goes to zero.
hence $$\hat f(\xi)=\frac{2}{1+\xi^2}\ \forall \ \xi\in \mathbb{R}$$
Is it correct answer?
Thanks in advance!