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I am wondering what is the inverse/opposite factorial function? e.g $\text{inverse-factorial}*(6)=3$

Furthermore, I am intrigued to know the answer to:

$$a!=\pi $$ Find $a$.

I would really appreciate if anyone could explain this to me as I have found nowhere online with a good explanation of inverse factorial functions. Thanks

TShiong
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Cameron Gray
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    See https://math.stackexchange.com/questions/931846/does-the-gamma-function-have-an-inverse and http://www.ams.org/journals/proc/2012-140-04/S0002-9939-2011-11023-2/ – lhf Nov 14 '18 at 12:21
  • Can you elaborate on what you want out of this? The most popular continuous version of factorial is the Gamma function, but I don't know if "ask a calculator to approximate the solution to $\Gamma(x)=\pi$ and add $1$ to the answer" is the sort of thing you're looking for. For detailed strategies for approximating answers, maybe see https://math.stackexchange.com/a/2739498/26369 – Mark S. Nov 14 '18 at 12:29
  • No integer's factorial is $pi$ the only thing you have seen is $\frac{1}{2}!=\pi$ which is not true. See definition of Gamma function : https://en.wikipedia.org/wiki/Gamma_function and see what happened when we put $\frac{1}{2}$ – Sujit Bhattacharyya Nov 14 '18 at 12:33
  • also refer to https://math.stackexchange.com/questions/1624347 – G Cab Nov 14 '18 at 12:53

2 Answers2

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Unfortunately there is not a closed form or nice series for the inverse of the factorial (or Gamma function).

First obstacle is that the factorial has a local minimum at $x:\;\psi(x)=0\; \to \; x=0.4616..$, so , considering only positive values of the argument, that gives you two values for the inverse.

For an analysis of the problem please refer to this and this papers.
A lighter look is given in this other paper.

Finally an interesting approximated function is given here.

G Cab
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inverse functions are not well defined when it is not a $1:1$ function, and as there is a minimum where: $$\Gamma'(z+1)=0$$ that is: $$\int_0^\infty\partial_nt^ze^{-t}dt=0$$ which can be numerically estimated.

Henry Lee
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