Function $n(s)$ is given implicitly as $n! = s$. How can I find an asymptotic for $n(s)$? I cannot understand, how can I make an explicit function from this to calculate an asymptotic, as there is no reverse operation for factorial (I mean, like root is reversal to pow, and so on).
Asymptotic for implicitly given function
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asymptotics
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0See [here](http://mathoverflow.net/questions/12828/inverse-gamma-function). – 2012-10-06
2 Answers
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Hint: Use the Stirling Approximation to the factorial. To get information about the growth of $n$ in terms of $s$, you will need asymptotic information about the Lambert function.
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From Stirling's approximation, you know that (for $n\to\infty$) $s(n) \sim \sqrt{2\pi n} \left( \frac{n}e \right)^n.$ The task is to invert this asymptotic relation.
Note that the dominant part reads $s\sim n^n$, i.e., $\log s \sim n \log n$ and $\log\log s \sim \log n + \log\log n$. As the first term dominates, we obtain the first approximation $n(s) \sim \log s.$
A better approximation reads $n \sim \frac{\log s}{\log\log s}$ and so on.