Inverse factorial function

Question

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

Answer 1

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.

Answer 2

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.