I am interested in the proof of the Inverse Fourier Transform for absolutely integrable real valued functions.
The proof I have read asks you to consider an auxiliary function $g_{a}(x)$ defined as
$$g_{a}(x) = \int_{-\infty}^{\infty} e^{-a \lvert t\rvert}e^{2\pi i t x}\operatorname{d}\!t = \frac{2a}{4\pi^{2}x^{2}+a^{2}}.$$
It then proceeds to prove that $\lim_{a\rightarrow 0 }f*g_{a}(x) = f(x)$ and therefore shows that $\hat{\hat{f}}(x) = f(-x)$ by the dominated convergence theorem.
My question is as follows: Where did $g$ come from?
So far, I've noticed that $g(x)$ is the form of a polynomial of degree two higher than the numberator, so perhaps there is a relationship between this expression and hyperbolic cotangent or the digamma function?
I've also noticed that $g_{a}(x)$ is a probability distribution function which looks a lot like the Cauchy Distribution. As a consequence to this, it feels like computing the Fourier transform of $g$ is going to be somewhat related to the Cauchy Distribution's characteristic function?
That being said, I have a feeling that I am just looking for similarities which are not useful at all. Can I gain any useful insights from those two observations? If not, how can I reasonably arrive at this integral equation for $g_{a}(t)$ "naturally"?
As for my meaning of "naturally", consider a standard epsilon-delta proof. When you let $\epsilon>0$ be given and consider $\delta = f(\epsilon)$, most of the intuition behind the proof itself was completely hidden by considering such a $\delta$ if you never show how you arrived at that $\delta$. In a similar vein, how do I reasonably arrive at $g_{a}(x)$?