0

Using a single function, how can I model a logarithmic-like increase followed by an exponential-like decay? The transition will occur at some critical value, Tcrit. A graph of my experimental data is given below. Tcrit is approximately 475.

enter image description here

antuasallong
  • 11
  • 2
  • Is your function allowed to have parameters? - which can change? – Sort of Damocles Oct 21 '19 at 15:33
  • Are "special" functions permitted? Some, like the Bessel functions, have a critical point such that to one side of a critical point, they oscillate about an equilibrium. On the other , they decay or increase exponentially. Other functions with such critical point might do it. – TurlocTheRed Oct 21 '19 at 15:33

1 Answers1

0

If you allow the sign function, I have a nice solution for you.

Let $\operatorname{sign}(x) = \begin{cases} 1 & x \geq 0 \\ -1 & x < 0 \end{cases}$

Choose $$f(x) = -\operatorname{sign}(x) \left(\alpha - e^{-\beta x + \operatorname{sign}(x)} \right) $$

where $\alpha$ must be chosen so that $\lim_{x\rightarrow 0^+} f(x) = \lim_{x\rightarrow 0^-} f(x)$, and $\beta$ controls the slope.

You can shift and scale from there. Here's what it looks like when $\alpha = 1.55$ and $\beta = 1.5$ (note that this value of $\alpha$ does not satisfy the condition I gave above).

example with alpha = 1.55 and beta = 1.5

Sort of Damocles
  • 2,443