Consider the function:

$$ f_{q}\left(x\right)=x+x^{q} $$

where $q\in\mathbb{Q},q>0$ and $x \in \mathbb{R}$, $x\geq 0$.

I am wondering what would be a method for inverting this function. It is monotone increasing on the non-negative reals as far as I see and so should be invertible.

Is there a closed form for the inverse function? Otherwise, could we express it as a series or an integral perhaps?

Thank you

  • 4
    $\begingroup$ It'n not something that you can do in general. $\endgroup$ – Surb Aug 13 '17 at 12:00
  • 1
    $\begingroup$ Of course you can do it in general. Puiseux series If $q=a/b$ with $(a,b)=1$ replace $x$ with $X^b$. The equation becomes $y=X^b+X^a$. Then you can expand $X$ as a Puiseux series. $\endgroup$ – uSir470888 Aug 13 '17 at 12:08
  • 1
    $\begingroup$ See the introduction of ... cs.uwaterloo.ca/research/tr/1993/03/W.pdf $\endgroup$ – Donald Splutterwit Aug 13 '17 at 12:56
  • $\begingroup$ For $q=a/b,a\le5,b\le5$, and $(a,b)=1$, the inverse can be explicitly written using regular radicals and Bring radicals. $\endgroup$ – Simply Beautiful Art Aug 13 '17 at 13:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.