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? enter image description here

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: enter image description here

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).enter image description here

  • $\begingroup$ 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. $\endgroup$
    – user120911
    Sep 11 '17 at 21:57

This a sigmoidal function on a finite interval that I have described previously here. The equation for the function is given by

$$f\left( x \right)=\frac{1}{2}\left[ 1+\text{sgn} (x)\,\frac{B\left( {1}/{2}\;,p+1,{{\left| x \right|}^{2}} \right)}{B\left( {1}/{2}\;,p+1 \right)} \right]$$

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.

Sigmoidal Fn Demo

  • $\begingroup$ you will notice in the edit of this question, I have tried to implement your Sigmoid Function in Mathematica, but with failure. Can you see what I am doing wrong? Also, how does your Sigmoid Function handle my specific case, where "the point of intersecting lines" can be shifted, and specifically, is shifted horizontally at (x, alpha / 2) while keeping the y-intercept = alpha and the x intercept = 0.5? $\endgroup$
    – user120911
    Sep 12 '17 at 7:01
  • $\begingroup$ @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! $\endgroup$ Sep 12 '17 at 13:21
  • $\begingroup$ @user120911 Have you resolved the beta function problem in Mathematica? (see previous comment) $\endgroup$ Sep 13 '17 at 20:42

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