# What is the most simple formula to achieve this pattern? (Part II)

This is a continuation of an earlier problem (What is the most simple formula to achieve this pattern?). In that question I assumed the point $(0.25, \frac{\alpha}{2})$ was fixed. But suppose this "point of intersection of curves" is more dynamic, yet in such a way that $\frac{\alpha}2$ still remains. For example $(0.4, \frac{\alpha}2)$. Does a formula exist, that captures this possibility while adhering to the prior constraints?

COMMENT: The following image shows my attempt at implementing Cye Waldman's answer (see below) in Mathematica. Obviously, something is wrong, but I can't see where my formula and Cye's formula diverge:

I have added another image to show exactly what the function should be able to capture. Note the possible "tuning" of the "point of intersecting lines" while keeping the asymptotic intersections with (0,alpha) and (0.5,0).

• Obviously, the straight line solution no longer applies when the "point of intersection of curves" shifts, given that (0,a) and (0.5,0) must be enforced. Sep 11 '17 at 21:57

$$f\left( x \right)=\frac{1}{2}\left[ 1+\text{sgn} (x)\,\frac{B\left( {1}/{2}\;,p+1,{{\left| x \right|}^{2}} \right)}{B\left( {1}/{2}\;,p+1 \right)} \right]$$
where the numerator and denominator $B$s are the incomplete and complete beta functions, respectively.
The figure below shows a range of the parameter space, $p$. Here, $p$ varies from $p=0$ (the straight line) to $p=20$ in increments of $2$. The $x$ and $y$ coordinates can be adjusted to match your requirements.
• @user120911 I follow the mathematical convention in the Atlas of Functions for $B(\nu,\mu,z)$ whereas Mathematica and Matlab use $B(z,\nu,\mu)$. Try that. Sorry for the long delay; must live in different time zones! Sep 12 '17 at 13:21