# A strange sigmoid-like function

I have come across the pattern shown below, and I have been searching in vain for functions that can describe what I see. It is not a standard sigmoid function, having local peaks before tapering off at either end. Also, as the steepness is reduced, these peaks become more attenuated.

Can anyone think of where they might have seen a function that captures this pattern, or perhaps someone has the imagination to think of one?

• The derivatives of $e^{-kx^2}$ for different (positive) $k$ looks something like this. Larger $k$ makes it more attenuated. – Arthur Sep 28 '17 at 7:36
• Looks similar to the Gibbs overshoot for the partial sums of a Fourier series approximating a discontinuous function. Although I think in that case the heights are preserved with finer and finer approximations. en.wikipedia.org/wiki/Gibbs_phenomenon – Michael Sep 28 '17 at 7:38
• Just out curioisity : how did you generate these curves ? – Claude Leibovici Sep 28 '17 at 7:56
• @Arthur Not sure if OP's function vanishes at infinity. Maybe add a real sigmoid-function like here: $\frac{\mathrm d}{\mathrm dx}[e^{-x^2}]+\arctan(5x)$. – M. Winter Sep 28 '17 at 8:08
• Following on from Arthur's comment, I remember having to sketch the derivative of the normal distribution curve a few years back, and it looks like this, except for a negative sign. Changing the variance can scale it to look more similar to what you have – Jihoon Kang Sep 28 '17 at 8:09

$$a\sin(b\tanh(cx))).$$