# Inverse Fourier Transform of the Cauchy distribution

Given the characteristic function of the cauchy distribution in the form

$$\hat{f}(q) = \exp(-\gamma|q|)$$

I am unsure how to derive the original probability distribution function

$$f(x) = \frac{\gamma}{\pi(\gamma^2+x^2)}$$

via the inverse Fourier transform, which I have tried using the following form.

$$f(x) = \frac{1}{2\pi}\int_{-\infty}^\infty \hat{f}(q)e^{iqx} \, dq$$

I suspect I am going wrong when transforming with respect to the absolute value $|q|$ in the characteristic function as I am unsure how to eliminate it. Any insight is very much appreciated.

Let the Cauchy distribution be $$\frac\gamma{\pi(\gamma^2+x^2)} \, dx$$ so that if $X$ is a random variable with that distribution then $$\Pr(X\in A) = \int_{-\infty}^\infty \frac \gamma{\pi(\gamma^2+x^2)} \, dx$$ for every Borel set $A.$
Then the characteristic function is $$q\mapsto \operatorname E\left( e^{iqX} \right) = \int_{-\infty}^\infty e^{iqx} \frac \gamma {\pi(\gamma^2+x^2)} \, dx.$$ Note that it says $e^{iqX},$ not $e^{-iqX}.$
An inversion theorem for this kind of transform will need a $\text{“}{-}\text{''}$ where you have a $\text{“}{+}\text{''}.$