I have the standard bump function below.
$$ \Psi(x) = e^{-\frac{1}{1 - \mathrm{min}(1, x^2)}} $$
How can I customize it to be like below:
Translate and scale
I can translate and scale by:
$$ \Psi(x) = e^{1-\frac{1}{1 - \mathrm{min}(1, (x-x_0)^2)}} $$
But I don't know how to deal with the $x_0 - \delta$ to $x_0 + \delta$ range.
Range
Is this the correct final answer?
$$ \Psi(x) = e^{1-\frac{\delta^2}{\delta^2 - \mathrm{min}(\delta^2, (x-x_0)^2)}} $$