Prove that there is no closed form of the inverse of the expression $y = x\cot \frac{\pi }{x}$

Question

Prove that there is no closed form of the inverse of the expression $y = x\cot \frac{\pi }{x}$ where $x \geq 3$. I am currently completely lost.

Answer 1

This is not really answer, just an explanation of why you're not likely to get a satisfactory answer.

This is just a guess, on my part, but proving that the inverse function can't be expressed in "closed form" (using other elementary functions) is somewhat analogous to the problem of proving that certain anti-derivatives cannot be expressed using elementary functions. This latter problem was studied by Liouville, and he proved that antiderivatives of certain functions (like $x^x$ and $e^{-x^2}$) can not be expressed via elementary functions. The proof uses some pretty sophisticated mathematical tools. See here for a bit more info.

Though the analogy is pretty fuzzy, my guess is that your problem is similarly difficult.