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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?

enter image description here

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    $\begingroup$ The derivatives of $e^{-kx^2}$ for different (positive) $k$ looks something like this. Larger $k$ makes it more attenuated. $\endgroup$ – Arthur Sep 28 '17 at 7:36
  • $\begingroup$ 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 $\endgroup$ – Michael Sep 28 '17 at 7:38
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    $\begingroup$ Just out curioisity : how did you generate these curves ? $\endgroup$ – Claude Leibovici Sep 28 '17 at 7:56
  • $\begingroup$ @Arthur Not sure if OP's function vanishes at infinity. Maybe add a real sigmoid-function like here: $\frac{\mathrm d}{\mathrm dx}[e^{-x^2}]+\arctan(5x)$. $\endgroup$ – M. Winter Sep 28 '17 at 8:08
  • $\begingroup$ 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 $\endgroup$ – Jihoon Kang Sep 28 '17 at 8:09
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You can take any "normal" sigmoid and compose it with a "folding function" (odd function, linear at the origin, with two extrema).

For instance

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

By playing with the parameters, you can achieve different effects.

enter image description here

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  • $\begingroup$ Yves, I am a little unsure what exactly you mean by "compose it with". Exactly how do you combine the two? Could you please post the equations that generated your chart? This looks very useful. $\endgroup$ – user120911 Sep 28 '17 at 10:16
  • $\begingroup$ @user120911: it is in front of your eyes. $\endgroup$ – Yves Daoust Sep 28 '17 at 10:20
  • $\begingroup$ Oh, I had the impression you took the formula you show above, and combine it (somehow) with a sigmoid function. My question was regarding the "somehow", but I guess the shown function is your solution, and it can create the patterns you demonstrate above. Is that correctly understood? $\endgroup$ – user120911 Sep 28 '17 at 10:30
  • $\begingroup$ Yes it is understood. $\endgroup$ – Yves Daoust Sep 28 '17 at 10:31

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