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I'm looking for a name for the following function

$$f(x) = |x|^n \cdot \operatorname{sgn}(x)$$

which is the exponent function, but with the sign stripped from the input and reattached to the output.

It seems useful enough for DSP that it ought to have a name, but Google's not much help with this sort of thing. Has anyone seen this function in any DSP software? What was it named?

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    $\begingroup$ is $n$ a non-negative integer? if so, i would call it a bipolar power function. if $n$ is odd, then $f(x)=x^n$ and is just a "power function". but when $n$ is even then then $|x|^n=x^n$ and is a uni-polar (always $\ge$ 0). so then the $\operatorname{sgn}(x)$ function makes it bipolar. if $n$ is any old real number, i dunno what i would call it. maybe the same. $\endgroup$ – robert bristow-johnson Jan 22 '18 at 6:11
  • $\begingroup$ Maybe you should rather describe your problem with that function than ask for its name! $\endgroup$ – Marcus Müller Jan 22 '18 at 6:49
  • $\begingroup$ hay @RodrigodeAzevdo, you should look at this and this. $\endgroup$ – robert bristow-johnson Jan 22 '18 at 20:02
  • $\begingroup$ @Guest : Please see this question and answer. I, as a mod, don't have rights to merge accounts; that has to be one via email with a real SE employee. You'll need to create a real account (if Guest isn't one) and let SE know which account that is. They'll then be able to verify that you're coming from the same IP address and probably other factors, to be sure they're merging you with the right account. $\endgroup$ – Peter K. Jan 22 '18 at 20:30
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There are many instances (eg Handbook of Computational and Numerical Methods in Finance, p. 42, Eq. 2.25) where it is simply called the "signed power" "signed power-$p$ function" or "signed power law".

Some instances are more common. The term "signed square-root" seems to be used, see for instance in this Simulink documentation:

Square root of the absolute value of the input, multiplied by the sign of the input.

with the signed square as an inverse. It is used in the realm of statistical distributions, contrast enhancement, or on data features (like SURF) as a compander, close in spirit to the A-law or the mu-law.

For parabolas (and even powers in general), the rewriting into $x\to x.|x|$ could be called a split parabola (see Selected solutions, HW 1, due January 22, 2009, an instance of split functions).

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