# Inverse factorial function

I am wondering what is the inverse/opposite factorial function? e.g inverse-factorial(6)=3

Furthermore, I am intrigued to know the answer to:

a!=π 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

• – 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 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 : 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 math.stackexchange.com/questions/1624347 – G Cab Nov 14 '18 at 12:53

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.
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.