I'm trying to modify an arctan function such that it asymptotes to some predefined y value at some predefined position along an x input array.
Does anyone know how to do this or can point me in the right direction?
So far I have been using:
modified arctan = $\frac{2}{\pi} \text{arctan} \left(\pi x \right)$ ,
To ensure that the function aymptotes to a y value of 1.
And I assume that one can manipulate the region at which it approaches 1 by changing:
$\text{arctan} \left(\pi x \right)$ to $\text{arctan} \left(\frac{\pi x}{N} \right)$
Obviously I realise that this function will never actually meet the required y value but perhaps there's a way to put a handle on where the curve reaches some predefined gradient... say 1 or $<0.5$ etc.
But where this handle N is, I'm not sure and maybe there's a better way to accomplish this.
Any help would be greatly appreciated so thanks in advance!