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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?

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    You can do such things with the Lambert W-function, otherwise no. http://en.wikipedia.org/wiki/Lambert_W_function2012-10-13

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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.