# inversion of the function $\sqrt x \ln x$

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 ?? thanks.

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You can do such things with the Lambert W-function, otherwise no. en.wikipedia.org/wiki/Lambert_W_function –  Will Jagy Oct 13 '12 at 22:08

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