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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 \(v=\Im(f)=\mathrm {const}\).


Also the notations


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,


should not be confused with the \(c\)th power of \(z!\), which is


Properties of ArcFactorial

ArcFactorial is holomorphic at the whole complex \(z\) plane except the half-line

\(z\le \mathrm{Homer}\)


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

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


Bart is the branchpoint;


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:


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:

\(\mathrm{ArcFactorial}(\mathrm{Homer}+t)=\mathrm{Bart} + \mathrm{Liza}_1 t^{1/2} + \mathrm{Liza}_2 t + \mathrm{Liza}_3 t^{3/2}+ ...\)


\(\mathrm{Liza}_1 =\sqrt{\frac{2}{\mathrm{Factorial}''(\mathrm{Bart})}} \approx ~ 1.5276760433847776\)
\(\mathrm{Liza}_2 \approx ~ 0.3559463008501492\)
\(\mathrm{Liza}_3 \approx \! -0.4620189870305121\)
\(\mathrm{Liza}_4 \approx \! -0.19468729558612438 \)



Factorial, Inverse function,