Arcfactorial

map of ArcFactorial

ArcFactorial is the principal branch of the inverse function of Factorial;

$ \mathrm{Factorial}(\mathrm{ArcFactorial}(z))=z$

Complex map of $f=\mathrm{ArcFactorial}(x\!+\!\mathrm i y)$ is shown at right in the $x$, $y$ plane with levels $u=\Re(f)=\mathrm {const}$ and levels levels $v=\Im(f)=\mathrm {const}$.

Notations

Also the notations

$\mathrm{afac}=\mathrm{ArcFactorial}=\mathrm{Factorial}^{-1}$

are suggested.

However, $\mathrm{Factorial}^{-1}(z)$ should not be confused with

$\displaystyle \mathrm{Factorial}(z)^{-1} = \frac{1}{\mathrm{Factorial}(z)}$

and, in general, the $c$th iteration of Factorial, id est,

$\mathrm{Factorial}^x(z)$

should not be confused with the $c$th power of $z!$, which is

$\mathrm{Factorial}(z)^c$

Properties of ArcFactorial

ArcFactorial is holomorphic at the whole complex $z$ plane except the half-line

$z\le \mathrm{Homer}$

where

$\mathrm{Homer}=\mathrm{Factorial}(\mathrm{Bart})\approx 0.8856031944108887$

and $\mathrm{Bart}\approx 0.4616321449683622$ is solution of equation

$\mathrm{Factorial}'(\mathrm{Bart})=0$

Bart is the branchpoint;

$\mathrm{ArcFactorial}(\mathrm{Homer})=\mathrm{Bart}$

At large values of argument, the ArcFactorial shows slow growth, similar to that of logarithm.

Real argument and special cases

$y=\mathrm{ArcFactorial}(x)$ and its asymptotic approximation

Behavior of ArcFactorial along the real axis is shown in figure at right.

For some natural values of argument, ArcFactorial has natural values:

$\mathrm{ArcFactorial}(1)=1$

$\mathrm{ArcFactorial}(2)=2$

$\mathrm{ArcFactorial}(6)=3$

$\mathrm{ArcFactorial}(24)\!=4$

and so on.

At certain specific values of argument, ArcFactorial has half-integer values:

$\displaystyle \mathrm{ArcFactorial}\left( \frac{\sqrt{\pi}}{2}\right)\!=\frac{1}{2}$

$\displaystyle \mathrm{ArcFactorial}\left( \frac{3\sqrt{\pi}}{4}\right)\!=\frac{3}{2}$

$\displaystyle \mathrm{ArcFactorial}\left( \frac{15\sqrt{\pi}}{8}\right)\!=\frac{5}{2}$

Expansion at Homer

The expansion of Factorial at its minimum has form

$\mathrm{Factorial}(\mathrm{Bart}+t)=\mathrm{Homer}

+\frac{1}{2}\mathrm{Factorial}(\mathrm{Bart}) t^2 +\frac{1}{6}\mathrm{Factorial}(\mathrm{Bart}) t^3+ .. $

The InverseSeries gives the expansion for the ArcFactorial:

$\mathrm{ArcFactorial}(\mathrm{Homer}+t)=\mathrm{Bart}

+ \mathrm{Liza}_1 t^{1/2} + \mathrm{Liza}_2 t + \mathrm{Liza}_3 t^{3/2}+ ...$ where

$\mathrm{Liza}_1 =\sqrt{\frac{2}{\mathrm{Factorial}(\mathrm{Bart})}} \approx ~ 1.5276760433847776$

$\mathrm{Liza}_2 \approx ~ 0.3559463008501492$

$\mathrm{Liza}_3 \approx \! -0.4620189870305121$

$\mathrm{Liza}_4 \approx \! -0.19468729558612438 $

Keywords

Factorial, Inverse function,