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?
Inversion of the function $ \sqrt x \ln x $
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logarithms
inverse
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0You can do such things with the Lambert W-function, otherwise no. http://en.wikipedia.org/wiki/Lambert_W_function – 2012-10-13
1 Answers
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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.