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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

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

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


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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