I was thinking about approximating the Gaussian curve with a triangular curve. The graphs look like this:
their respective functions are:
$$ y_1(x) = t(x) = max(0, 1 - |x|)$$
$$ y_2(x) = e^{ - x^2}$$
where the domain $x \in \mathbb{R}$.
I wanted to do something (potentially crazy) and use that approximation (or something of that sort that made this work) but with complex numbers.
So the idea is if I have a complex number $ z = re^{i \theta} \in \mathbb{C}$ to (somehow) use the approximation:
$$ e^{-|x|^2} = \phi(x) \approx t(x) = max(0, 1 - |x|) \approx re^{i \theta} = z$$
So the intuition (that might be completely wrong) is that because complex numbers can be expressed using the exponential function, maybe can they be expressed using $ e^{-|x|^2}$ and hence, some for of the triangle function?