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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.

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I fear your question may change after this answer, but I'll take a shot at it. (I.e. might you correct a mistake and then go one to find that there are others? After all, you're not specific about details of how you did this.)

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 $\text{“}{+}\text{''}.$

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