Use characteristic function of X. (X is a Laplace (0,1) distribution) to obtain characteristic function of the standar Cauchy distribution

Question

Let X a r.v. with pdf: $\;f(x) = \tfrac{1}{2}e^{-|x|}$ (Laplace(0,1))

a) Calculate the characteristic function of X

No problem. I do it. $\varphi_{X}(t)=\tfrac{1}{1+t{^2}}$

b) Use the previous result to obtain the characteristic function of the standard Cauchy distribution

I know Cauchy standar pdf is $\tfrac{1}{\pi}\varphi_{X}(t)$, but I don't know how to proceed.

Answer 1

This is an immediate application of the inversion theorem for Fourier transforms. See https://en.wikipedia.org/wiki/Fourier_inversion_theorem