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Is there a term for a smooth $\mathbb R\to\mathbb R^+$ function that is unimodal (has one local maximum), and whose $n^{\text{th}}$ derivative has $(n+1)$ local extrema? An example of such a function is the Gaussian function $f(x)=\exp(-x^2)$.

Plot of the Gaussian function and its derivatives

A negative example of a unimodal function that does not fit these criteria is $f(x)=1/(10+x^4)$. Its derivatives have too many extrema.

Plot of the rational function and its derivatives

Another negative example that shows that the number of extrema can grow exponentially (with respect to the order of a derivative) is the Rvachev function $f(x)=\operatorname{up}\left(x/4\right)$.

Plot of the Rvachev function and its derivatives

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  • $\begingroup$ If there is no existing term exactly matching my description, I would appreciate if somebody pointed me to some closely related classes of functions. $\endgroup$ Apr 2, 2020 at 22:08

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I am sure better answers exist, and they would be most welcome indeed!

For now, since no one has been able to answer yet, and the user has now written that if there's no answer, they would like to know a "closely related classes of functions", I am providing my best attempt below.

Unimodal functions whose $n^{\rm{th}}$ derivative has $(n+1)$ local extrema, eigenfunctions of a Morse potential:

enter image description here

The lowest function ($v=0$) is unimodal, and the the $v^{\rm{th}}$ function has $v+1$ local extrema.

The "harmonic oscillator" can be considered an approximation to the Morse potential (as in the image below):

enter image description here

The $v=0$ eigenfunction of the harmonic oscillator potential is precisely Gaussian (like the function in the title of your question), however the $v=0$ eigenfunction of the Morse potential is slightly different due to the anharmonicity in the potential.

The $v=$ eigenfunction of the Morse function is given by $\Psi_n(z)$:

enter image description here

where $n=v=0$, $\lambda$ is a positive constant, and $x$ and $x_e$ denote the variable on the abscissa axis and its value at the global minimum of the Morse potential respectively.


Therefore, a closely related function to the kind you describe, is the generalized Laguerre polynomial given here, in the $n=0$ case. The lowest eigenfunction $\Psi_0(z)$ of a Morse potential, might be a sub-class of the final answer you're seeking, and a super-class of the functions you have already proposed (Gaussians).

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The Gaussian term looks as a single solution.

In particular, from an analysis of the parameters of the Rodrigues formula for classical orthogonal polynomials, only a Gaussian base function allows an additional factor in the form of a constant.

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    $\begingroup$ Doesn't $1/(1+x^2)$ also satisfy the requirements? $\endgroup$ Apr 9, 2020 at 2:16

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