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,