Inversion of the function $ \sqrt x \ln x $

Question

Is there an exact (not asymptotic) inversion of the function $ \sqrt x \ln x $ or can we only obtain this inverse in terms of a power series?

Answer 1

Let $x = e^z$. We have: $$ y = \sqrt{x} \ln{x} = \exp\left(\frac{z}{2}\right) z $$

Thus: $$ \frac{y}{2} = \frac{z}{2} \exp\left(\frac{z}{2}\right) $$

Using the Lambert W-function, we have: $$ \frac{z}{2} = W\left(\frac{y}{2}\right) $$

Put $x$ back to get: $$ x = \exp\left(2 W\left(\frac{y}{2}\right)\right) $$

This is as close to a closed form as you can get. The function cannot be expressed in elementary functions.