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Is the functional inverse of any elementary function asymptotic to some elementary function ? For instance Lambert's $W(z)$ is asymptotic to $ln(z)$. See http://mathworld.wolfram.com/LambertW-Function.html

Another example is the inverse of $z^5 + z$ ~ $z^{\dfrac{1}{5}}$.

Also the inverse of $-(ln(2 sinh(z)/z) - z)$ ~ $e^z$

etc.

My motivation is based on induction ideas. A much weaker question is if this is true for the inverse of elementary meromorphic functions.

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Very interesting.

I think you can prove it recursively, but it will be a little long, so I won't do it here.

First, note that this is true for all basic elementary functions (polynomials, $\exp$, $\tanh$, ...). You have to find for each one an asymptote of their reverse function ($x^{\frac{1}{degree}}$ for polynomials, $\ln$ for $\exp$, ...).

Then, for any way of building an elementary function (unary and binary), find an elementary asymptote as well.

Example :

  • Let us consider $f$ an injective elementary function whose inverse $f^{-1}$ has for asymptote $a$, which is elementary. Then, $f^n $ is also elementary and injective. Let us note $g = (f^n)^{-1} $. Prove $g$ is asymptotic to $a^{\frac{1}{n}}$, or some other elementary function you build with $a$

  • Let us consider $f$ and $g$ two injective elementary functions whose inverses $f^{-1}$ and $g^{-1}$ have for asymptote respectively $a$ and $b$, which are elementary. If $\lim\limits_{x\rightarrow \infty} \frac{f(x)}{g(x)} = 0$, then (I think it is $a$, but prove it) $a$ is asymptotic to $(f+g)^{-1}$, if $\lim\limits_{x\rightarrow \infty} \frac{f(x)}{g(x)} = \infty$, then ...

  • and so on

Once you have done all these cases, you are done.

If you cannot prove one step (but I doubt it will happen), it may highlight a case for which it isn't true. But still, you will know it to be true for many cases of elementary functions

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