Yet another request for how to do the Inverse (Gamma(x-1) Function -- involves Lambert W

Question

I have looked far and wide here and at other math forums for how to compute the Inverse of the Gamma Function as it relates to Factorials, ie: getting x from x!

So far, these two links seems to be to hold the most promise for what I'm looking for: Inverse of a factorial and David Cantrell's posting: Link

robjohn and Claude Leibovici posted good explanations as to what Cantrell was trying to say.

$$ n=[e^{W(log[n!/√{2π}]/e)+1}−1/2] $$

However, I'm running into the problem in that I CAN'T figure out how to solve the portion of the formula that involves the Lambert W Function!

I can get up to this point: $s=(log[n!/√{2π}]/e)$, but I can't for the life of me figure out how to proceed with $n=[e^{W(s)+1}−1/2]$. And unfortunately, the Wikipedia article for the Lambert W is sorely lacking on how to actually apply it to solving problems involving it. Can someone please help? Thanks.

Answer 1

I just found a video by BlackRedPen on youtube talking about the W Lambert: https://www.youtube.com/watch?v=sWgNCra93D8 where he’s telling us how to solve for $x^x=2$. I still don’t have a math proof to answer my question on how to use Lambert W for Cantrell’s work, but BlackRedPen does finish off his lecture with the answer in the form of $x=e^{W(ln2)}$… then he points the audience to Wolfram’s “Productlog” function! Plugging in what I worked Cantrell’s formula out to be above, I get:

In terms of Γ(x), it shows up as: $[e^{W(log[n!/√{2π}]/e)+1}−1/2]$, here;

if you want in terms of n!, change the formula to $[e^{W(log[n!/√{2π}]/e)+1}+1/2]$, here

which works out pretty close when using Windows Calc.exe.

I hope that helps anyone else who might be beating their head against the wall as I have had. And if anyone has an elegant expression that answers my previous post, I would be extremely grateful.