Solve $x = e^{x^\alpha}$.

Question

I'd like to solve an equation of the form $x = e^{x^\alpha}$, for a given $\alpha$. I saw that the Lambert W function can help to solve some equations, but it seems useless when $\alpha \neq 1$.

Thank you!

Answer 1

HINT:

Exponentuate both sides by $\alpha$: $$\frac{\alpha x^\alpha}{\alpha}=e^{\alpha x^\alpha}$$

Let $u=\alpha x^{\alpha}$, $$\frac{u}\alpha=e^u$$ which can be solved by Lambert W function easily: $$-ue^{-u}=-\alpha$$ $$-u=W(-\alpha)$$

ADDED: To make the answer complete, $$\alpha x^{\alpha}=-W(-\alpha)$$ $$x=\left(-\frac{W(-\alpha)}\alpha\right)^{1/\alpha}$$

It seems like that the expression inside the big brackets are always positive as long as $W$ produces real values, and thus $x$ is real for $\alpha\le\frac1e$.

ADDED 2: It might be interesting to consider the limit $\alpha\to0$. I believe $x=e$ in this limit, but am not able to prove it...