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The function $$f(x )= x \operatorname e^{|x|} $$has an interesting semi-sigmoid shape. However, it is somehow a "horizontal" sigmoid. I would like to know the function of the vertical equivalent. So what is the inverse of this function?

Sewer Keeper
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Holgersson Måns
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    The Wikipedia plot summary for the Lambert W function may be relevant here. – Brian Tung Jun 03 '20 at 15:46
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    Notice that $f(x)> 0$ for $x>0$—and it is a odd function—so it suffices to consider the inverse of the function $(0, \infty) \to (0, \infty)$ – Good Boy Jun 03 '20 at 16:00
  • Thanks for your input so far. @GoodBoy; yes, it should suffice to find the inverse of ()=e() – Holgersson Måns Jun 03 '20 at 16:07
  • As written by Brian Rung, the function you are looking for is called Lambert W function. It is a special function. You can not express it with other simpler functions, but you can find plot and properties on Wikipedia. – LL 3.14 Jun 03 '20 at 17:32
  • Thank you all! Now that I have learned that the function has a name a whole new world has opened up for me :-). I have already found a bunch om interesting articles dealing with the function. Good to know that there is no inverse of it (the effort I put in trying to find one almost drove me crazy...). Seems as if it is a quite an established function; albeit not in my area (statistical models of psychological phenomena). I can already see som nice applications of the function in my area :-) – Holgersson Måns Jun 03 '20 at 19:13
  • So the inverse of ()=()exp(x) is called the "Lambert W"-function. Then; what is the original function called? I suppose there is a better name than just "the inverse of Lambert W". If not; can anyone come up with a better name than "the xexpabs-function"? – Holgersson Måns Jun 04 '20 at 11:25
  • According to Goerg a possible application of the (x)exp(x)-function might be when we want to "Gaussianize" skewed distributions of data in order to be "allowed" to analyze the distributions with parametric statistical tests. See for example datahttps://www.ncbi.nlm.nih.gov/pmc/articles/PMC4562338/ I have a hunch that the (x)exp(x)-function will do a better job than the logit-function when we want to ”Gaussianize” proportion data (or probabities or percentages) . However, whether that is true remains to be seen… I will explore it further. – Holgersson Måns Jun 04 '20 at 11:53

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