Difference between revisions of "ArcFactorial"

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m (Text replacement - "\$([^\$]+)\$" to "\\(\1\\)")
 
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[[File:AfacmapT800.png|300px|right|thumb|map of ArcFactorial]]
 
[[File:AfacmapT800.png|300px|right|thumb|map of ArcFactorial]]
 
'''ArcFactorial''' is the principal branch of the inverse function of [[Factorial]];
 
'''ArcFactorial''' is the principal branch of the inverse function of [[Factorial]];
: $ \mathrm{Factorial}(\mathrm{ArcFactorial}(z))=z$
+
: \( \mathrm{Factorial}(\mathrm{ArcFactorial}(z))=z\)
   
[[Complex map]] of $f=\mathrm{ArcFactorial}(x\!+\!\mathrm i y)$ is shown at right in the $x$, $y$ plane with
+
[[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 \(u=\Re(f)=\mathrm {const}\) and levels
levels $v=\Im(f)=\mathrm {const}$.
+
levels \(v=\Im(f)=\mathrm {const}\).
   
 
==Notations==
 
==Notations==
   
 
Also the notations
 
Also the notations
: $\mathrm{afac}=\mathrm{ArcFactorial}=\mathrm{Factorial}^{-1}$
+
: \(\mathrm{afac}=\mathrm{ArcFactorial}=\mathrm{Factorial}^{-1}\)
 
are suggested.
 
are suggested.
   
However, $\mathrm{Factorial}^{-1}(z)$ should not be confused with
+
However, \(\mathrm{Factorial}^{-1}(z)\) should not be confused with
: $\displaystyle \mathrm{Factorial}(z)^{-1} = \frac{1}{\mathrm{Factorial}(z)}$
+
: \(\displaystyle \mathrm{Factorial}(z)^{-1} = \frac{1}{\mathrm{Factorial}(z)}\)
   
and, in general, the $c$th iteration of Factorial, id est,
+
and, in general, the \(c\)th iteration of Factorial, id est,
: $\mathrm{Factorial}^x(z)$
+
: \(\mathrm{Factorial}^x(z)\)
should not be confused with the $c$th power of $z!$, which is
+
should not be confused with the \(c\)th power of \(z!\), which is
: $\mathrm{Factorial}(z)^c$
+
: \(\mathrm{Factorial}(z)^c\)
   
 
==Properties of ArcFactorial==
 
==Properties of ArcFactorial==
ArcFactorial is holomorphic at the whole complex $z$ plane except the half-line
+
ArcFactorial is holomorphic at the whole complex \(z\) plane except the half-line
: $z\le \mathrm{Homer}$
+
: \(z\le \mathrm{Homer}\)
 
where
 
where
: $\mathrm{Homer}=\mathrm{Factorial}(\mathrm{Bart})\approx 0.8856031944108887$
+
: \(\mathrm{Homer}=\mathrm{Factorial}(\mathrm{Bart})\approx 0.8856031944108887\)
and $\mathrm{Bart}\approx 0.4616321449683622$
+
and \(\mathrm{Bart}\approx 0.4616321449683622\)
 
is solution of equation
 
is solution of equation
: $\mathrm{Factorial}'(\mathrm{Bart})=0$
+
: \(\mathrm{Factorial}'(\mathrm{Bart})=0\)
   
 
Bart is the [[branchpoint]];
 
Bart is the [[branchpoint]];
:$\mathrm{ArcFactorial}(\mathrm{Homer})=\mathrm{Bart}$
+
:\(\mathrm{ArcFactorial}(\mathrm{Homer})=\mathrm{Bart}\)
   
 
At large values of argument, the ArcFactorial shows slow growth, similar to that of [[logarithm]].
 
At large values of argument, the ArcFactorial shows slow growth, similar to that of [[logarithm]].
   
 
==Real argument and special cases==
 
==Real argument and special cases==
[[File:AfacplotT2px300.png|600px|right|thumb|$y=\mathrm{ArcFactorial}(x)$ and its asymptotic approximation]]
+
[[File:AfacplotT2px300.png|600px|right|thumb|\(y=\mathrm{ArcFactorial}(x)\) and its asymptotic approximation]]
 
Behavior of ArcFactorial along the real axis is shown in figure at right.
 
Behavior of ArcFactorial along the real axis is shown in figure at right.
   
 
For some [[natural number|natural]] values of argument, ArcFactorial has natural values:
 
For some [[natural number|natural]] values of argument, ArcFactorial has natural values:
: $\mathrm{ArcFactorial}(1)=1$
+
: \(\mathrm{ArcFactorial}(1)=1\)
: $\mathrm{ArcFactorial}(2)=2$
+
: \(\mathrm{ArcFactorial}(2)=2\)
: $\mathrm{ArcFactorial}(6)=3$
+
: \(\mathrm{ArcFactorial}(6)=3\)
: $\mathrm{ArcFactorial}(24)\!=4$
+
: \(\mathrm{ArcFactorial}(24)\!=4\)
 
and so on.
 
and so on.
   
 
At certain specific values of argument, ArcFactorial has half-integer values:
 
At certain specific values of argument, ArcFactorial has half-integer values:
: $\displaystyle \mathrm{ArcFactorial}\left( \frac{\sqrt{\pi}}{2}\right)\!=\frac{1}{2}$
+
: \(\displaystyle \mathrm{ArcFactorial}\left( \frac{\sqrt{\pi}}{2}\right)\!=\frac{1}{2}\)
: $\displaystyle \mathrm{ArcFactorial}\left( \frac{3\sqrt{\pi}}{4}\right)\!=\frac{3}{2}$
+
: \(\displaystyle \mathrm{ArcFactorial}\left( \frac{3\sqrt{\pi}}{4}\right)\!=\frac{3}{2}\)
: $\displaystyle \mathrm{ArcFactorial}\left( \frac{15\sqrt{\pi}}{8}\right)\!=\frac{5}{2}$
+
: \(\displaystyle \mathrm{ArcFactorial}\left( \frac{15\sqrt{\pi}}{8}\right)\!=\frac{5}{2}\)
   
 
==Expansion at Homer==
 
==Expansion at Homer==
 
The expansion of [[Factorial]] at its minimum has form
 
The expansion of [[Factorial]] at its minimum has form
   
: $\mathrm{Factorial}(\mathrm{Bart}+t)=\mathrm{Homer}
+
: \(\mathrm{Factorial}(\mathrm{Bart}+t)=\mathrm{Homer}
 
+\frac{1}{2}\mathrm{Factorial''}(\mathrm{Bart}) t^2
 
+\frac{1}{2}\mathrm{Factorial''}(\mathrm{Bart}) t^2
+\frac{1}{6}\mathrm{Factorial'''}(\mathrm{Bart}) t^3+ .. $
+
+\frac{1}{6}\mathrm{Factorial'''}(\mathrm{Bart}) t^3+ .. \)
   
 
The [[InverseSeries]] gives the expansion for the ArcFactorial:
 
The [[InverseSeries]] gives the expansion for the ArcFactorial:
   
: $\mathrm{ArcFactorial}(\mathrm{Homer}+t)=\mathrm{Bart}
+
: \(\mathrm{ArcFactorial}(\mathrm{Homer}+t)=\mathrm{Bart}
 
+ \mathrm{Liza}_1 t^{1/2}
 
+ \mathrm{Liza}_1 t^{1/2}
 
+ \mathrm{Liza}_2 t
 
+ \mathrm{Liza}_2 t
+ \mathrm{Liza}_3 t^{3/2}+ ...$
+
+ \mathrm{Liza}_3 t^{3/2}+ ...\)
 
where
 
where
: $\mathrm{Liza}_1 =\sqrt{\frac{2}{\mathrm{Factorial}''(\mathrm{Bart})}} \approx ~ 1.5276760433847776$
+
: \(\mathrm{Liza}_1 =\sqrt{\frac{2}{\mathrm{Factorial}''(\mathrm{Bart})}} \approx ~ 1.5276760433847776\)
: $\mathrm{Liza}_2 \approx ~ 0.3559463008501492$
+
: \(\mathrm{Liza}_2 \approx ~ 0.3559463008501492\)
: $\mathrm{Liza}_3 \approx \! -0.4620189870305121$
+
: \(\mathrm{Liza}_3 \approx \! -0.4620189870305121\)
: $\mathrm{Liza}_4 \approx \! -0.19468729558612438 $
+
: \(\mathrm{Liza}_4 \approx \! -0.19468729558612438 \)
   
 
==References==
 
==References==

Latest revision as of 18:26, 30 July 2019

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

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

References


Keywords

Factorial, Inverse function,