Difference between revisions of "Factorial"

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<div class="thumb tright"><div style="width:7em;">
  
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{|class="wikitable" style="margin:0; text-align: right;" cellspacing="0"
  
{|class="wikitable" style="margin:0; text-align: right;" cellspacing="0"
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! \(n\)
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! <math>n</math>
  
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! <math>n!</math>
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! \(n!\)
  +
|-
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| \(-5/2\) || \(4\sqrt{\pi}/3\)
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|-
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| \(-2\) || \(\infty\)
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|-
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| \(-3/2\) || -2\(\sqrt{\pi}\)
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|-
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| \(-1\) || \(\infty\)
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|-
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| \(-1/2\) ||\(\sqrt{\pi}\)
 
|-
  
|-
 
| 0 || 1
  
| 0 || 1
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|-
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| 1/2 ||\(\sqrt{\pi}/2\)
 
|-
  
|-
 
| 1 || 1
  
| 1 || 1
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|-
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| 3/2 ||\(3 \sqrt{\pi}/4\)
 
|-
  
|-
 
| 2 || 2
  
| 2 || 2
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|-
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| 5/2 ||\(15 \sqrt{\pi}/8\)
 
|-
  
|-
 
| 3 || 6
  
| 3 || 6
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| 10 || 3628800
  
| 10 || 3628800
 
|}
  
|}
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In [[mathematics]], the '''factorial''' is the [[meromorphic function]] with fast growth along the real axis; for non-negative integer values of the argument, this function has integer values. <ref name="IRENE">
 
  +
Warning: This article is under construciton. It still has mistakes.
  +
  +
In [[mathematics]], [[Factorial]] is [[meromorphic function]] with fast growth along the real axis; for non-negative integer values of the argument, this function has integer values. <ref name="IRENE">
 
http://people.math.sfu.ca/~cbm/aands/page_255.htm
  
http://people.math.sfu.ca/~cbm/aands/page_255.htm
 
M.Abramowitz and I.Stegun: Handbook of Mathematical Functions. 6. Gamma Function and Related Functions (2010)
  
M.Abramowitz and I.Stegun: Handbook of Mathematical Functions. 6. Gamma Function and Related Functions (2010)
 
</ref>
  
</ref>
−
Frequently, the [[postfix]] notation <math>n!</math> is used for the factorial of number <math>n</math>.
 
−
For integer <math>n</math>, the <math>n!</math> gives the number of ways in which ''n'' labelled objects (for example the numbers from 1 to ''n'') can be arranged in order. These are the [[permutation]]s of the set of objects.
 
−
In some [[programming language]]s, both n! and factorial(n) , or Factorial(n), are recognized as the factorial of the number <math>n</math>.
  
   
  +
Frequently, the [[postfix]] notation \(n!\) is used for the factorial of number \(n\).
−
<!--The ${\rm factorial}(z)$ can be considered also as [[meromorphic]] function of complex variable $z$. !-->
  
  +
For integer \(n\), the \(n!\) gives the number of ways in which ''n'' labelled objects (for example the numbers from 1 to ''n'') can be arranged in order. These are the [[permutation]]s of the set of objects.
−
The factorial, applied $m$ times is called $m$th iteration of factorial, it can be denoted as ${\rm factorial}^m(z)$;
  
  +
−
In particular, ${\rm factorial}^2(z)={\rm factorial}\big( {\rm factorial}(z)\big)$ and ${\rm factorial}^{-1}$ is inverse function of factorial; in some range of values of $z$, the relation ${\rm factorial}^{-1}\big( {\rm factorial}(z) \big)=z$.
  
  +
In some [[programming language]]s, both n! and factorial(n) , or Factorial(n), are recognized as the factorial of the number \(n\).
−
Such iteration can be defined also for non-integer values of $m$, number of iterations can be complex <ref name="qf">
  
−
http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1 D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.</ref>.
  
−
In particular, at $m=1/2$, we have
 
−
${\rm factorial}^{1/2}(z)=\sqrt{\rm factorial}(z)$, where function $\sqrt{\,!\,}=\sqrt{\rm factorial}$ can be interpreted as [[square root of factorial]]: $\sqrt{\,!\,}\Big(\sqrt{\,!\,}(z)\Big)=z!$. This function is used as logo of the [[Physics department of the Moscow State University]] and part of logo of [[Main Page|TORI]] shown in the top left corner.
  
   
 
==Integer values of the argument==
  
==Integer values of the argument==
 
For integer values of the argument, the factorial can be defined by a [[recurrence relation]]. If ''n'' labelled objects have to be assigned to ''n'' places, then the ''n''-th object can be placed in one of ''n'' places: the remaining ''n''-1 objects then have to be placed in the remaining ''n''-1 places, and this is the same problem for the smaller set. So we have
  
For integer values of the argument, the factorial can be defined by a [[recurrence relation]]. If ''n'' labelled objects have to be assigned to ''n'' places, then the ''n''-th object can be placed in one of ''n'' places: the remaining ''n''-1 objects then have to be placed in the remaining ''n''-1 places, and this is the same problem for the smaller set. So we have
   
−
:<math> n! = n \cdot (n-1)! \,</math>
 +
:\( n! = n \cdot (n-1)! \,\)
   
 
and it follows that
 
and it follows that
   
−
:<math> n! = n \cdot (n-1) \cdots 2 \cdot 1 , \,</math>
 +
:\( n! = n \cdot (n-1) \cdots 2 \cdot 1 , \,\)
   
 
which we could derive directly by noting that the first element can be placed in ''n'' ways, the second in ''n''-1 ways, and so on until the last element can be placed in only one remaining way.
  
which we could derive directly by noting that the first element can be placed in ''n'' ways, the second in ''n''-1 ways, and so on until the last element can be placed in only one remaining way.
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The factorial function is found in many combinatorial counting problems. For example, the [[binomial coefficient]]s, which count the number of subsets size ''r'' drawn from a set of ''n'' objects, can be expressed as
  
The factorial function is found in many combinatorial counting problems. For example, the [[binomial coefficient]]s, which count the number of subsets size ''r'' drawn from a set of ''n'' objects, can be expressed as
   
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:<math>\binom{n}{r} = \frac{n!}{r! (n-r)!} .</math>
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:\(\binom{n}{r} = \frac{n!}{r! (n-r)!} .\)
   
 
The factorial function can be extended to arguments other than positive integers: this gives rise to the [[Gamma function]].
  
The factorial function can be extended to arguments other than positive integers: this gives rise to the [[Gamma function]].
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===Implicit definition===
  
===Implicit definition===
−
The factorial can be defined as unique meromorphic function <math>F</math>, satisfying
+
The factorial can be defined as unique meromorphic function \(F\), satisfying
 
relations
  
relations
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:<math> F(z+1)=(z+1) F(z) </math>
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:\( F(z+1)=(z+1) F(z) \)
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:<math> F(0)=1 </math>
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:\( F(0)=1 \)
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for all complex <math>z</math> except negative integer values.
+
for all complex \(z\) except negative integer values.
−
The [[uniqueness]] of function <math>F</math>, satisfying these equations, follows from the
+
The [[uniqueness]] of function \(F\), satisfying these equations, follows from the
 
[[Wielandt's theorem]]
  
[[Wielandt's theorem]]
 
<ref name="Wielandt's theorem">
  
<ref name="Wielandt's theorem">
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===Definition through the integral===
  
===Definition through the integral===
−
Usually, the integral representation is used as definition. For <math>\Re(z)>-1</math>, define
 +
Usually, the integral representation is used as definition. For \(\Re(z)>-1\), define
−
:<math> z! = \int_0^\infty t^z \exp(-t) \mathrm{d}t </math>
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:\( z! = \int_0^\infty t^z \exp(-t) \mathrm{d}t \)
   
 
Such definition is similar to that of the [[Gamma function]], and leads to the relation
  
Such definition is similar to that of the [[Gamma function]], and leads to the relation
−
:<math>z!=\Gamma(z+1)</math>
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:\(z!=\Gamma(z+1)\)
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for all complex <math>z</math> except the negative integer values.
+
for all complex \(z\) except the negative integer values.
   
−
The definition above agrees with the combinatoric definition for integer values of the argument; at integer <math>z</math>, the integral can be expressed in terms of the elementary functions.
+
The definition above agrees with the combinatoric definition for integer values of the argument; at integer \(z\), the integral can be expressed in terms of the elementary functions.
   
 
===Extension of integral definition===
  
===Extension of integral definition===
−
<!--
 +
<!-- ??
−
{{Image|Factorialz.jpg|right|400px|<math>f=z!</math> in the complex <math>z</math>-plane.}}
 +
{{fig|Factorialz.jpg|400|0|2|8|\(f=z!\) in the complex \(z\)-plane.}}
 
!-->
  
!-->
  +
<div class="thumb tright" style="float:right; margin:-24px 0px 2px 8px">
−
[[File:Factorialz.jpg|right|400px|thumb|<math>f=z!</math> in the complex <math>z</math>-plane.]]
  
  +
{{pic|Factorialz.jpg|400px}}<small><center>\(f=z!\) in the complex \(z\)-plane.
  +
</center></small></div>
   
 
The definition through the integral can be extended to the whole complex plane, using relation
 
The definition through the integral can be extended to the whole complex plane, using relation
−
:<math>z!=(z+1)!/(z+1)</math>
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:\(z!=(z+1)!/(z+1)\)
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for the cases <math>\Re(z)<-1</math>, assuming that <math>z</math> is not negative integer.
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for the cases \(\Re(z)<-1\), assuming that \(z\) is not negative integer.
−
Also, the symmetry formula takes place for the non-integer values of <math>z</math>,
 +
Also, the symmetry formula takes place for the non-integer values of \(z\),
−
:<math>z! (-z)! =\frac{\pi z}{\sin(\pi z)}</math>
 +
:\(z! (-z)! =\frac{\pi z}{\sin(\pi z)}\)
 
The similar formula of symmetry holds however, for the [[Gamma function]].
  
The similar formula of symmetry holds however, for the [[Gamma function]].
−
From this expression, it follows that <math>\frac{1}{z!(-z)!}</math> is [[entire function]] of <math>z</math>.
+
From this expression, it follows that \(\frac{1}{z!(-z)!}\) is [[entire function]] of \(z\).
−
Also, this symmetry gives the simple way to express <math>(1/2)!=\sqrt{\pi}/2</math>, and, therefore, factorial of half-integer numbers.
 +
Also, this symmetry gives the simple way to express \((1/2)!=\sqrt{\pi}/2\), and, therefore, factorial of half-integer numbers.
   
 
<!-- The f allows to plot the map of factorial in the complex plane.
 
<!-- The f allows to plot the map of factorial in the complex plane.
 
!-->
  
!-->
 
In the figure,
 
In the figure,
−
lines of constant <math>u=\Re(z!)</math> and
+
lines of constant \(u=\Re(z!)\) and
−
lines of constant <math>v=\Im(z!)</math> are shown.<br>
 +
lines of constant \(v=\Im(z!)\) are shown.<br>
  +
The levels
−
The levels u = − 24, − 20, − 16, − 12, − 8, − 7, − 6, − 5, − 4, − 3, − 2, − 1,0,1,2,3,4,5,6,7,8,12,16,20,24 are drown with thick black lines.<br>
  
  +
\(u = − 24,\) \(− 20,\) \(− 16,\) \(− 12,\) \(− 8,\) \(− 7,\) \(− 6, \) \(− 5,\) \(− 4,\) \(− 3,\) \(− 2,\) \(− 1,\) \(
  +
0,\) \(1,\) \(2,\) \(3,\) \(4,\) \(5,\) \(6,\) \(7,\) \(8,\) \(12,\) \(16,\) \(20,\) \(24\) are drown with thick black lines.<br>
 
Some of intermediate levels u = const are shown with thin blue lines for positive values and with thin red lines for negative values.
  
Some of intermediate levels u = const are shown with thin blue lines for positive values and with thin red lines for negative values.
 
<br>
  
<br>
−
The levels v = − 24, − 20, − 16, − 12, − 8, − 7, − 6, − 5, − 4, − 3, − 2, − 1 are shown with thick red lines.<br>
 +
Levels \(v = − 24,\) \(− 20,\) \(− 16,\) \(− 12,\) \(− 8,\) \( − 7,\) \( − 6,\) \( − 5,\) \( − 4,\) \( − 3,\) \( − 2,\) \( − 1\) are shown with thick red lines.<br>
−
The level v = 0 is shown with thick pink lines.<br>
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Level \(v = 0\) is shown with thick pink lines.<br>
  +
Levels \(v = 1,\) \(
−
The levels v = 1,2,3,4,5,6,7,8,12,16,20,24 are drown with thick blue lines. some of intermediate levels v = const are shown with thin green lines.<br>
  
  +
2,\) \(
−
The dashed blue line shows the level <math>u=\mu_0</math> and corresponds to the value <math>\mu_0=(x_0)!\approx 0.85</math> of the principal local minimum <math>x_0\approx 0.45</math> of the factorial of the real argument.<br>
  
  +
3,\) \(
  +
4,\) \(
  +
5,\) \(
  +
6,\) \(
  +
7,\) \(
  +
8,\) \(
  +
12,\) \(
  +
16,\) \(
  +
20,\) \(
  +
24~\)
  +
are drown with thick blue lines. some of intermediate levels \(v = \mathrm{const}\) are shown with thin green lines.<br>
  +
The dashed blue line shows the level \(u=\mu_0\) and corresponds to the value \(\mu_0=(x_0)!\approx 0.85\) of the principal local minimum \(x_0\approx 0.45\) of the factorial of the real argument.<br>
 
The dashed red line shows the level and corresponds to the similar value of the negative local extremum of the factorial of the real argument.<br>
  
The dashed red line shows the level and corresponds to the similar value of the negative local extremum of the factorial of the real argument.<br>
 
Due to the fast growth of the function, in the right hand side of the figure, the density of the levels exceeds the ability of the plotter to draw them; so, this part is left empty.
  
Due to the fast growth of the function, in the right hand side of the figure, the density of the levels exceeds the ability of the plotter to draw them; so, this part is left empty.
   
 
==Factorial of the real argument==
  
==Factorial of the real argument==
  +
<div class="thumb tright" style="float:right; margin:-18px 0px 2px 8px; width:360px">
−
<!--{{Image|FactoReal.jpg|right|400px|
  
−
<math>\mathrm{factorial}(x)=x!</math>; the inverse function, id est,
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{{pic|FactoReal.jpg|320px}}<small><center>\(\mathrm{factorial}(x)=x!\); the inverse function, i.e.,
−
<math>\mathrm{factorial}^{-1}(x)</math>, and
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\(\mathrm{factorial}^{-1}(x)\), and \(\mathrm{factorial}(x)^{-1}\) versus real \(x\).</center></small>
  +
</div>
−
<math>\mathrm{factorial}(x)^{-1}</math>
 
  +
The definition above was elaborated for factorial of complex argument. In particular, it can be used to evlauate the factorial of the real argument. In the figure at right, the \(\mathrm{factorial}(x)=x!\) is plotted versus real \(x\) with red line. The function has simple poluses at negative integer \(x\).
−
versus real <math>x</math>.}} !-->
  
−
[[File:FactoReal.jpg|right|440px|thumb|
  
−
$\mathrm{factorial}(x)=x!$; the inverse function, i.e.,
 
−
$\mathrm{factorial}^{-1}(x)$, and
  
−
$\mathrm{factorial}(x)^{-1}$ versus real $x$.]]
  
−
The definition above was elaborated for factorial of complex argument. In particular, it can be used to evlauate the factorial of the real argument. In the figure at right, the <math>\mathrm{factorial}(x)=x!</math> is plotted versus real <math>x</math> with red line. The function has simple poluses at negative integer <math>x</math>.
  
   
−
At <math>x\rightarrow -1+o</math>, the <math>x! \rightarrow +\infty</math>.
 +
At \(x\rightarrow -1+o\), the \(x! \rightarrow +\infty\).
 
===The local minimum===
  
===The local minimum===
 
The factorial has local minimum at
 
The factorial has local minimum at
−
: <math>x=\nu_0\approx 0.4616321450</math>
 +
: \(x=\nu_0\approx 0.4616321450\)
−
<!--: <math>x=\nu_0\approx 0.461632144968362341262659542325721328468196204</math>!-->
 +
<!--: \(x=\nu_0\approx 0.461632144968362341262659542325721328468196204\)!-->
 
marked in the picture with pink vertical line; at this point, the derivative of the factorial is zero:
  
marked in the picture with pink vertical line; at this point, the derivative of the factorial is zero:
−
:<math>\mathrm{factorial}^{\prime}(\nu_0)=0</math>
 +
:\(\mathrm{factorial}^{\prime}(\nu_0)=0\)
 
The value of factorial in this point
  
The value of factorial in this point
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: <math>\mu_0=\nu_0!=\mathrm{factorial(\nu_0)}\approx 0.8856031944</math>
 +
: \(\mu_0=\nu_0!=\mathrm{factorial(\nu_0)}\approx 0.8856031944\)
−
<!-- <math>\mu_0=\nu_0!=\mathrm{factorial(\nu_0)}\approx 0.88560319441088870027881590058258873320795153367 </math>!-->
 +
<!-- \(\mu_0=\nu_0!=\mathrm{factorial(\nu_0)}\approx 0.88560319441088870027881590058258873320795153367 \)!-->
−
The Tailor expansion of <math>z!</math> at the point <math>z=\nu_0</math> can be writen ax follows:
 +
The Tailor expansion of \(z!\) at the point \(z=\nu_0\) can be writen ax follows:
−
:<math>z!=\mu_0+\sum_{n=2}^{N-1} c_n (z-\nu_0)^n + \mathcal{O}(z-\nu_0)^N~</math> .
 +
:\(z!=\mu_0+\sum_{n=2}^{N-1} c_n (z-\nu_0)^n + \mathcal{O}(z-\nu_0)^N~\) .
 
The coefficients of this expansion are copypasted in the table below:
  
The coefficients of this expansion are copypasted in the table below:
 
{|class="wikitable" style="margin:0; text-align: right;" cellspacing="0"
  
{|class="wikitable" style="margin:0; text-align: right;" cellspacing="0"
  +
! \(n\)
−
! <math>n</math>
  
−
! approximation of <math>c_n</math>
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! approximation of \(c_n\)
 
|-
  
|-
 
| 2 || 0.428486815855585429730209907810650582960483696962
  
| 2 || 0.428486815855585429730209907810650582960483696962
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This expansion can be used for the precise evaluation of the inverse function of factorial (arcfactorial) in vicinity of the branchpoint.
  
This expansion can be used for the precise evaluation of the inverse function of factorial (arcfactorial) in vicinity of the branchpoint.
   
−
Other local extremums are at negative values of the argument; one of them in shown in the figure above.
 +
Other local extremums are at negative values of the argument; one of them in shown in the figure above. There, it is denoted as \(\nu_1\).
   
 
<!--
  
<!--
 
===Specific values===
  
===Specific values===
 
For several specific values of the argument, the simple representations for the factorial are known. In addition fo the integer values,
  
For several specific values of the argument, the simple representations for the factorial are known. In addition fo the integer values,
−
<math>\left(-\frac{1}{2}\right)!=\sqrt{\pi}</math>; then, using the relation<math>z!=z\cdot(z+1)!</math>, values at half-integer argument can be expressed; for example, <math>\left(\frac{1}{2}\right)!=\frac{\sqrt{\pi}}{2}\approx 0.8862269255</math> is slightly greater than <math>\mu_0</math>, which is minimal value of this function for the positive values of the argument.
 +
\(\left(-\frac{1}{2}\right)!=\sqrt{\pi}\); then, using the relation\(z!=z\cdot(z+1)!\), values at half-integer argument can be expressed; for example, \(\left(\frac{1}{2}\right)!=\frac{\sqrt{\pi}}{2}\approx 0.8862269255\) is slightly greater than \(\mu_0\), which is minimal value of this function for the positive values of the argument.
   
 
Values of factorial at integer plus quarter and at integer plut third are mentioned in in textbooks, but these numbers do not have special names.
  
Values of factorial at integer plus quarter and at integer plut third are mentioned in in textbooks, but these numbers do not have special names.
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==The Taylor expansion==
  
==The Taylor expansion==
−
The [[Taylor expansion]] of <math>z!</math> at <math>z=0</math>, or the [[MacLaurin expansion]], has the form
 +
The [[Taylor expansion]] of \(z!\) at \(z=0\), or the [[MacLaurin expansion]], has the form
−
<math>z!=\sum_{n=0}^{N-1} g_n z^n+\mathcal{O}(z^N)~</math> . The first coefficients of this expansion are copypasted in the table below:
 +
\(z!=\sum_{n=0}^{N-1} g_n z^n+\mathcal{O}(z^N)~\) . The first coefficients of this expansion are copypasted in the table below:
 
<table border="2" cellpadding="4" cellspacing="2" style="margin: 1em 1em 1em 0; background: #ffffff; border: 1px #aaa solid; border-collapse: collapse;">
  
<table border="2" cellpadding="4" cellspacing="2" style="margin: 1em 1em 1em 0; background: #ffffff; border: 1px #aaa solid; border-collapse: collapse;">
−
<tr><th><math>n</math></th> <th><math>g_n</math></th> <th>approximation of <math>g_n</math></th> </tr>
 +
<tr><th>\(n\)</th> <th>\(g_n\)</th> <th>approximation of \(g_n\)</th> </tr>
−
<tr><td>0</td> <td><math>1</math></td>
 +
<tr><td>0</td> <td>\(1\)</td>
  +
<td>\(\ 1.000000000000000000000000000000000000000000000000000000\)</td> </tr>
−
<td><math> 1.000000000000000000000000000000000000000000000000000000000000</math></td> </tr>
  
−
<tr><td>1</td><td><math>~-\gamma~</math></td>
 +
<tr><td>1</td><td>\(~-\gamma~\)</td>
  +
<td>\(-0.5772156649015328606065120900824024310421593359399235988\)</td> </tr>
−
<td><math>-0.577215664901532860606512090082402431042159335939923598805767</math></td> </tr>
  
−
<tr><td>2</td> <td><math>\frac{\pi^2}{12}+\frac{\gamma^2}{2}</math></td>
 +
<tr><td>2</td> <td>\(\frac{\pi^2}{12}+\frac{\gamma^2}{2}\)</td>
  +
<td>\(\ 0.9890559953279725553953956515006347079391835207282140904\)</td> </tr>
−
<td><math> 0.989055995327972555395395651500634707939183520728214090443192</math></td> </tr>
  
−
<tr><td>3</td><td><math>-\frac{\zeta(3)}{3}-\frac{\pi^2\gamma}{12}+\frac{\gamma^3}{6}</math></td>
 +
<tr><td>3</td><td>\(-\frac{\zeta(3)}{3}-\frac{\pi^2\gamma}{12}+\frac{\gamma^3}{6}\)</td>
  +
<td>\(-0.9074790760808862890165601673562751149286114490725637609\)</td></tr>
−
<td><math>-0.907479076080886289016560167356275114928611449072563760941328</math></td></tr>
  
−
<tr><td>4</td><td><math>\frac{\pi^4}{160}+\frac{\zeta(3)}{3}-\frac{\pi^2\gamma^2}{12}+\frac{\gamma^4}{24}</math></td>
 +
<tr><td>4</td><td>\(\frac{\pi^4}{160}+\frac{\zeta(3)}{3}-\frac{\pi^2\gamma^2}{12}+\frac{\gamma^4}{24}\)</td>
  +
<td>\(\ 0.9817280868344001873363802940218508503605736797234654154\)</td></tr>
−
<td><math> 0.981728086834400187336380294021850850360573679723465415404953</math></td></tr>
  
 
</table>
  
</table>
−
Here, <math>\gamma</math> is the [[Euler–Mascheroni constant|Euler constant]] and <math>\zeta</math> is the [[Riemann function]].
+
Here, \(\gamma\) is the [[Euler–Mascheroni constant|Euler constant]] and \(\zeta\) is the [[Riemann function]].
 
The [[Computer algebra system]]s such as [[Maple (sotfware)|Maple]] and [[Mathematica]] can generate many terms of this expansion.
  
The [[Computer algebra system]]s such as [[Maple (sotfware)|Maple]] and [[Mathematica]] can generate many terms of this expansion.
−
The [[radius of convergence]] of the Taylor series is unity, and the coefficient <math>g_n</math> does not decay as <math>n</math> increases. However, due to the relation <math>z!=z\cdot \mathrm{factorial}(z)</math>, for any real value of argument of factorial, the expansion above can be used for the precize evaluation of factorial of the real argument, running the approximation above for an argument with modulus not larget than halh.
 +
The [[radius of convergence]] of the Taylor series is unity, and the coefficient \(g_n\) does not decay as \(n\) increases. However, due to the relation \(z!=z\cdot \mathrm{factorial}(z)\), for any real value of argument of factorial, the expansion above can be used for the precize evaluation of factorial of the real argument, running the approximation above for an argument with modulus not larget than halh.
 
Also, the expansion at the half-integer values can be used:
  
Also, the expansion at the half-integer values can be used:
−
<math>z!=\sum_{n=0}^{N-1} h_n~\left(z-\frac{1}{2}\right)^n+\mathcal{O}\left(\left(z-\frac{1}{2}\right)^N\right)~</math> . The first coefficients of this expansion are copypasted in the table below:
 +
\(z!=\sum_{n=0}^{N-1} h_n~\left(z-\frac{1}{2}\right)^n+\mathcal{O}\left(\left(z-\frac{1}{2}\right)^N\right)~\) . The first coefficients of this expansion are copypasted in the table below:
 
<table border="2" cellpadding="4" cellspacing="2" style="margin: 1em 1em 1em 0; background: #ffffff; border: 1px #aaa solid; border-collapse: collapse;">
  
<table border="2" cellpadding="4" cellspacing="2" style="margin: 1em 1em 1em 0; background: #ffffff; border: 1px #aaa solid; border-collapse: collapse;">
−
<tr><th><math>n</math></th> <th><math>h_n</math></th> <th>approximation of <math>g_n</math></th> </tr>
 +
<tr><th>\(n\)</th> <th>\(h_n\)</th> <th>approximation of \(g_n\)</th> </tr>
−
<tr><td>0</td> <td><math>\frac{\sqrt{\pi}}{2}</math></td>
 +
<tr><td>0</td> <td>\(\frac{\sqrt{\pi}}{2}\)</td>
  +
<td>\( 0.88622692545275801364908374167057259139877472806 \)</td> </tr>
−
<td><math>0.886226925452758013649083741670572591398774728061193564106905 </math></td> </tr>
  
−
<tr><td>1</td><td><math>\frac{-\sqrt{\pi}}{2}(\gamma+\log(4)-2)</math></td>
 +
<tr><td>1</td><td>\(\frac{-\sqrt{\pi}}{2}(\gamma+\log(4)-2)\)</td>
  +
<td>\( 0.03233839744888501382886988426897030778133478887\)</td> </tr>
−
<td><math>0.0323383974488850138288698842689703077813347888705070206366386</math></td> </tr>
  
−
<tr><td>2</td> <td><math>\frac{\pi^{5/2}}{8}+\sqrt{\pi}\gamma -\sqrt{\pi}\log(4) +\frac{\gamma^2\sqrt{\pi}}{4} +\sqrt{\pi}\log(2)^2</math></td>
 +
<tr><td>2</td> <td>\(\frac{\pi^{5/2}}{8}+\sqrt{\pi}\gamma -\sqrt{\pi}\log(4) +\frac{\gamma^2\sqrt{\pi}}{4} +\sqrt{\pi}\log(2)^2\)</td>
  +
<td>\( 0.41481345368830116823003762311135634284890996337\)</td> </tr>
−
<td><math> 0.414813453688301168230037623111356342848909963370422367977736</math></td> </tr>
  
 
<tr><td>3</td><td> long expression</td>
  
<tr><td>3</td><td> long expression</td>
  +
<td>\(-0.10729480456477221168754195638970966205457592382\)</td></tr>
−
<td><math>-0.107294804564772211687541956389709662054575923821298300938631</math></td></tr>
  
 
<tr><td>4</td><td>even longer expression</td>
  
<tr><td>4</td><td>even longer expression</td>
  +
<td>\( 0.14464535904462154303833221025388452407002686153\)</td></tr>
−
<td><math>0.144645359044621543038332210253884524070026861530981428414028</math></td></tr>
  
 
</table>
  
</table>
−
The Taylor series of <math>z!</math> developed at <math>z=1/2</math>, converges for all <math>z</math> such that <math>|z-1/2|<3/2</math>.
+
The Taylor series of \(z!\) developed at \(z=1/2\), converges for all \(z\) such that \(|z-1/2|<3/2\).
   
−
In order to boost the approximation of factorial for real values of the argument, and, especially, for the evaluation for complex values, and the evaluation of the inverse function, the expansions for <math>\log(z!)</math>
 +
In order to boost the approximation of factorial for real values of the argument, and, especially, for the evaluation for complex values, and the evaluation of the inverse function, the expansions for \(\log(z!)\)
−
and <math>1/z!</math> are used instead of the direct Taylor expansions above.
 +
and \(1/z!\) are used instead of the direct Taylor expansions above.
   
 
==Halphing of the imaginary part of the argument==
  
==Halphing of the imaginary part of the argument==
−
While neither <math>z</math> nor <math>(z+1)/2</math> is negative integer, the argument of factorial can be dropped with the identity
 +
While neither \(z\) nor \((z+1)/2\) is negative integer, the argument of factorial can be dropped with the identity
  +
:\(
−
:<math>
  
 
\mathrm{Factorial}(z)~=~\frac{2^z}{\sqrt{\pi}}~
  
\mathrm{Factorial}(z)~=~\frac{2^z}{\sqrt{\pi}}~
 
\mathrm{Factorial}\!\left(\frac{z}{2} \right)\cdot
  
\mathrm{Factorial}\!\left(\frac{z}{2} \right)\cdot
 
\mathrm{Factorial}\!\left(\frac{z\!-\!1}{2} \right)
  
\mathrm{Factorial}\!\left(\frac{z\!-\!1}{2} \right)
  +
\)
−
</math>
  
 
This may be useful to drop with factor 1/2 the imaginary part of the argument (for example, to apply the Taylor expansion above for the evaluation), but the function has to be evaluated twice.
  
This may be useful to drop with factor 1/2 the imaginary part of the argument (for example, to apply the Taylor expansion above for the evaluation), but the function has to be evaluated twice.
   
 
==Related functions==
  
==Related functions==
 
<!--{{Image|ArcGamma.jpg|right|400px|ArcFactorial in the complex plane.}}!-->
  
<!--{{Image|ArcGamma.jpg|right|400px|ArcFactorial in the complex plane.}}!-->
−
[[Image:ArcGamma.jpg|right|400px|thumb|ArcFactorial in the complex plane.]]
 +
{{fig|ArcGamma.jpg|360|-38|2|8|ArcFactorial in the complex plane.}}
   
 
In the plot of factorial of the real argument, the two other functions are plotted,
 
In the plot of factorial of the real argument, the two other functions are plotted,
−
<math>\mathrm{ArcFactorial}(x)=\mathrm{factorial}^{-1}(x)</math> and <math>\mathrm{factorial}(x)^{-1}</math>.
 +
\(\mathrm{ArcFactorial}(x)=\mathrm{factorial}^{-1}(x)\) and \(\mathrm{factorial}(x)^{-1}\).
 
These functions are used for the generalization of the factorial and for its evaluation.
  
These functions are used for the generalization of the factorial and for its evaluation.
 
===Inverse function===
  
===Inverse function===
−
<div class="thumb tright">
 +
<div class="thumb tright" style="float:right;">
−
<div style="width:19em;">
  
 
{|class="wikitable" style="margin:0; text-align: right;" cellspacing="0"
  
{|class="wikitable" style="margin:0; text-align: right;" cellspacing="0"
  +
! \(z\)
−
! <math>z</math>
  
−
! ArcFactorial<math>(z)</math>
 +
! ArcFactorial\((z)\)
 
|-
  
|-
 
<!--
  
<!--
 
| 0.88560319441088870027881590058258873320795153367 || 0.461632144968362341262659542325721328468196204
  
| 0.88560319441088870027881590058258873320795153367 || 0.461632144968362341262659542325721328468196204
 
|- !-->
  
|- !-->
−
| <math>\mu_0</math>
+
| \(\mu_0\)
 
|| 0.46163214496836234126265954233
  
|| 0.46163214496836234126265954233
 
|-
  
|-
−
| <math>\frac{\sqrt{\pi}}{2}</math>
+
| \(\frac{\sqrt{\pi}}{2}\)
 
|| 0.50000000000000000000000000000
  
|| 0.50000000000000000000000000000
 
|-
  
|-
Line 256: Line 280:
 
|-
  
|-
 
|}
  
|}
−
</div>
  
 
</div>
  
</div>
   
 
Inverse function of factorial can be defined with equation
  
Inverse function of factorial can be defined with equation
−
:<math>(\mathrm{ArcFactorial}(z))!=z</math>
 +
:\((\mathrm{ArcFactorial}(z))!=z\)
−
and condition that [[ArcFactorial]] is [[holomorphic]] in the comlex plane with cut along the part of the real axis, that begins at the minimum value of the factorial of the positive argument, and extends to <math>-\infty</math>. This function is shown with
+
and condition that [[ArcFactorial]] is [[holomorphic]] in the comlex plane with cut along the part of the real axis, that begins at the minimum value of the factorial of the positive argument, and extends to \(-\infty\). This function is shown with
−
lines of constant real part <math>u=\Re(\mathrm{ArcFactorial}(z))</math> and
+
lines of constant real part \(u=\Re(\mathrm{ArcFactorial}(z))\) and
−
lines of constant imaginary part <math>v=\Im(\mathrm{ArcFactorial}(z))</math>.
+
lines of constant imaginary part \(v=\Im(\mathrm{ArcFactorial}(z))\).
   
−
Levels <math>u=1,2,3</math> are shown with thick black curves.<br>
 +
Levels \(u=1,2,3\) are shown with thick black curves.<br>
−
Levels <math>u=
 +
Levels \(u=
−
0.2,0.4,0.6,0.8,
+
0.2,\) \(0.4,\) \(0.6,\) \(0.8,\) \(
−
1.2,1.4,1.6,1.8,
 +
1.2,\) \(1.4,\) \(1.6,\) \(1.8,\) \(
−
2.2,2.4,2.6,2.8,
 +
2.2,\) \(2.4,\) \(2.6,\) \(2.8,
−
3.2,3.4,3.6 </math> are shown with thin blue curves.<br>
 +
3.2,\) \(3.4,\) \(3.6\) are shown with thin blue curves.<br>
−
Levels <math>v=1,2,3</math> are shown with thick blue curves.<br>
 +
Levels \(v=1,2,3\) are shown with thick blue curves.<br>
−
Level <math>v=0</math> is shown with thick pink line.<br>
 +
Level \(v=0\) is shown with thick pink line.<br>
−
Levels <math>v=-1,-2,-3</math> are shown with thick red curves.<br>
 +
Levels \(v=-1,-2,-3\) are shown with thick red curves.<br>
−
The intermediate levels of constant <math>v</math> are shown with thin dark green curves.
 +
The intermediate levels of constant \(v\) are shown with thin dark green curves.
−
The ArcFactorial has the [[branch point]] <math>\mu_0 \approx 0.85 </math>; the cut of the range of holomorphizm is shown with black dashed line.
 +
The ArcFactorial has the [[branch point]] \(\mu_0 \approx 0.85 \); the cut of the range of holomorphizm is shown with black dashed line.
 
<!-- The figure shows the mapping ot the [[complex plane]] with the factorial function.
 
<!-- The figure shows the mapping ot the [[complex plane]] with the factorial function.
 
!-->
  
!-->
Line 281: Line 304:
 
ArcFactorial for some real values of the argument is approximated in the table at right.
  
ArcFactorial for some real values of the argument is approximated in the table at right.
   
−
<div class="thumb tright">
 +
<div class="thumb tright" style="width:19em;">
−
<div style="width:19em;">
  
 
{|class="wikitable" style="margin:0; text-align: right;" cellspacing="0"
  
{|class="wikitable" style="margin:0; text-align: right;" cellspacing="0"
  +
! \(n\)
−
! <math>n</math>
  
−
! approximation for <math>d_n</math>
 +
! approximation for \(d_n\)
 
|-
  
|-
 
| 1 || 1.43764234228440800
  
| 1 || 1.43764234228440800
Line 297: Line 319:
 
|}
  
|}
 
</div>
  
</div>
  +
In vicinity of the branchpoint \(z=\mu_0\), the ArcFactorial can be expanded as follows:
−
</div>
  
  +
:\(
−
In vicinity of the branchpoint <math>z=\mu_0</math>, the ArcFactorial can be expanded as follows:
  
−
:<math>
  
 
\mathrm{ArcFactorial}(z)=\nu_0+\sum_{n=1}^{N-1} d_n\cdot \Big(\log(z/\mu_0) \Big)^{n/2}
  
\mathrm{ArcFactorial}(z)=\nu_0+\sum_{n=1}^{N-1} d_n\cdot \Big(\log(z/\mu_0) \Big)^{n/2}
  +
\)
−
</math>
  
 
The approximations for the first coefficients of this expansion are copypasted in the table at right.
  
The approximations for the first coefficients of this expansion are copypasted in the table at right.
 
About of 30 terms in this expansion are sufficient to plot the distribution of the real and the imaginary parts of
 
About of 30 terms in this expansion are sufficient to plot the distribution of the real and the imaginary parts of
Line 310: Line 331:
 
<!-- In particular, factorial maps the unity to unity; two is mapped to two, and 3 is mapped to 6. !-->
  
<!-- In particular, factorial maps the unity to unity; two is mapped to two, and 3 is mapped to 6. !-->
   
−
===Function <math>f(z)=1/z!</math>===
 +
===Function f(z)=1/z! ===
  +
<div class="thumb tright" style="float:right; margin:2px 0px 4px 8px">
−
<!--{{Image|OneOverFactorial.jpg|right|400px|<math>f(z)=\frac{1}{z!}</math> in the complex <math>z</math>-plane.}} !-->
  
−
[[File:OneOverFactorial.jpg|right|400px|thumb|<math>f(z)=\frac{1}{z!}</math> in the complex <math>z</math>-plane.]]
 +
{{pic|OneOverFactorial.jpg|400px}}<small><center>\(f(z)=\frac{1}{z!}~\) in the complex \(z\)-plane.</center></small>
  +
</div>
−
The inverse function of factorial, id est, <math>\mathrm{ArcFactorial}(z)=\mathrm{Factorial}^{-1}(z)</math> from the previous section, sohuld not be confused with
  
  +
The inverse function of factorial, id est, \(\mathrm{ArcFactorial}(z)=\mathrm{Factorial}^{-1}(z)~\) from the previous section, sohuld not be confused with
−
<math>f(z)=\frac{1}{z!}=\mathrm{Factorial}(z)^{-1}=\frac{1}{\mathrm{Factorial}(z)}</math>
  
  +
−
shown in the figure at right.
 
  +
\(f(z)=\) \(\frac{1}{z!}\) \(
−
The lines of constant <math>u=\Re(f(z))</math> and
  
  +
=\mathrm{Factorial}(z)^{-1}\)
−
the lines of constant <math>v=\Im(f(z))</math> are drawn.<br>
  
  +
\(=\frac{1}{\mathrm{Factorial}(z)}~\),
−
The levels <math>u=-24,-20,-16,-12,-8,-7 .. 7,8,12,16,20,24</math> are shown with thick black lines.<br>
  
  +
which is also shown in the figure at right.
−
The levels <math>v=-24,-20,-16,-12,-8,-7 ... 7,-1</math> are shown with thick red lines.<br>
  
  +
The lines of constant \(u=\Re(f(z))\) and
−
The level <math>v=0</math> is shown with thick pink line.<br>
  
  +
the lines of constant \(v=\Im(f(z))\) are drawn.<br>
−
The levels <math>v=1,2, ... 7,8,12,16,20,24</math> are shown with thick blue lines.<br>
  
  +
The levels \(u=-24,\) \(-20,\) \(-16,\) \(-12,\) \(-8,\) \(-7\) \( .. 7,\) \(8,\) \(12,\) \(16,\) \(20,\) \(24\)
−
Some of intermediate elvels <math>u=</math>const are shown with thin red lines for negative values and thin blue lines for the positive values.<br>
  
−
Some of intermediate elvels <math>v=</math>const are shown with thin green lines.<br>
 +
are shown with thick black lines.<br>
  +
The levels \(v=-24,-20,-16,-12,-8,-7 ... 7,-1\) are shown with thick red lines.<br>
−
The blue dashed curves represent the level <math>u=1/\mu_0</math> and correspond to the positive local maximum of the inverse function of the real argument.<br>
  
  +
The level \(v=0\) is shown with thick pink line.<br>
−
The red dashed curves represent the level <math>u=1/\mu_1</math>, which corresponds to the first negative local maximum of the factorial of the real argument;
 
  +
The levels \(v=1,2, ... 7,8,12,16,20,24\) are shown with thick blue lines.<br>
−
<math>\mathrm{factorial}^{\prime}(\nu_1)=0</math>;
 
  +
Some of intermediate elvels \(u=\)const are shown with thin red lines for negative values and thin blue lines for the positive values.<br>
−
<math>\mathrm{factorial}(\nu_1)=\mu_1</math>.<br>
  
  +
Some of intermediate elvels \(v=\)const are shown with thin green lines.<br>
  +
The blue dashed curves represent the level \(u=1/\mu_0\) and correspond to the positive local maximum of the inverse function of the real argument.<br>
  +
The red dashed curves represent the level \(u=1/\mu_1\), which corresponds to the first negative local maximum of the factorial of the real argument;
  +
\(\mathrm{factorial}^{\prime}(\nu_1)=0\);
  +
\(\mathrm{factorial}(\nu_1)=\mu_1\).<br>
   
 
In the upper-left hand side of the figure, and at the lower-left hand side of the figure, the density of levels exceeds the ability of the ploter to draw them, and these parts are left empty.
  
In the upper-left hand side of the figure, and at the lower-left hand side of the figure, the density of levels exceeds the ability of the ploter to draw them, and these parts are left empty.
   
−
<math>f(z)=\frac{1}{z!}</math> is [[entire function]], that grows in the left hand side of the compelx plane and quickly decays to zero along the real axis.
 +
\(f(z)=\frac{1}{z!}\) is [[entire function]], that grows in the left hand side of the compelx plane and quickly decays to zero along the real axis.
   
 
<!--
  
<!--
 
===Evaluation of the factorial===
  
===Evaluation of the factorial===
−
In principle, the integral representation from the definition above can be used for the evlauation of the factorial. However, such an implementation is not efficient, and is not suitable, when the factorial is used as a component in construction of other functions with complicated representations, involving many evaluations of the factorial. Therefore, the approximations with elementary functions are used. For the efficient evaluation of <math>z!</math>, the approximatons for <math>1/(z!)</math> above, of those for <math>\log(z!)</math> are used.
 +
In principle, the integral representation from the definition above can be used for the evlauation of the factorial. However, such an implementation is not efficient, and is not suitable, when the factorial is used as a component in construction of other functions with complicated representations, involving many evaluations of the factorial. Therefore, the approximations with elementary functions are used. For the efficient evaluation of \(z!\), the approximatons for \(1/(z!)\) above, of those for \(\log(z!)\) are used.
 
!-->
  
!-->
   
  +
== [[LogFactorial]] ==
−
===Logfactorial===
  
 
<!--{{Image|LogFactorialZ.jpg|right|400px|LogFactorial at the complex plane.}} !-->
  
<!--{{Image|LogFactorialZ.jpg|right|400px|LogFactorial at the complex plane.}} !-->
  +
<div class="thumb tright" style="float:right; margin: 0px 0px 2px 8px">
−
[[File:LogFactorialZ.jpg|right|400px|thumb|LogFactorial at the complex plane]]
  
  +
{{pic|LogFactorialZ.jpg|400px}}<small><center>LogFactorial at the complex plane</center></small>
  +
</div>
   
 
For the approximation of factorial, it can be expressed as
  
For the approximation of factorial, it can be expressed as
−
: <math> z!=\exp(\mathrm{LogFactorial}(z)) </math>
 +
\[ z!=\exp(\mathrm{LogFactorial}(z))
  +
\]
 
At large values of the argument, the LogFactorial does not show the fast growth and is easier for the numerical implementation, than factorial.
  
At large values of the argument, the LogFactorial does not show the fast growth and is easier for the numerical implementation, than factorial.
  +
−
====Plof of LogFactorial====
  
  +
At real positive values of the argument, [[LogFactorial]] can be interpreted as [[logarithm]] of [[Factorial]]:
−
Function LogFactorial is shown in figure with lines of constant real part and lines of constant imaginary part.
  
  +
\[
−
Levels of constant <math>u=\Re(\mathrm{LogFactorial}(z))</math> and
  
−
Levels of constant <math>v=\Im(\mathrm{LogFactorial}(z))</math> are drawn with solid lines:<br>
 +
\mathrm{LogFactorial}(x)=\ln(\mathrm{LogFactorial}(x)), \ x>0
  +
\]
−
Levels <math>u=-24,-20,-16,-12,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,12,16,20,24</math> are shown with thick black curves. <br>
  
  +
−
Levels <math>u=
  
  +
Function [[LogFactorial]] is shown in figure a right with lines of constant real part and lines of constant imaginary part.
−
-3.8,-3.6,-3.4,-3.2
  
  +
Levels of constant \(u=\Re(\mathrm{LogFactorial}(z))\) and
−
-2.8,-2.6,-2.4,-2.2
  
  +
Levels of constant \(v=\Im(\mathrm{LogFactorial}(z))\) are drawn with solid lines:<br>
−
-1.8,-1.6,-1.4,-1.2
  
  +
Levels \(u=-24,\) \(-20,\) \(-16,\) \(-12,\) \(-8,\) \(-7,\) \(-6,\) \(-5,\) \(-4,\) \(-3,\) \(-2,\) \(-1,\) \(0,\) \(1,\) \(2,\) \(3,\) \(4,\) \(5,\) \(6,\) \(7,\) \(8,\) \(12\) \(,16,\) \(20,\) \(24\) are shown with thick black curves. <br>
−
-0.8,-0.6,-0.4,-0.2</math> are shown with thin red curves.<br>
  
−
Levels <math>u=
 +
Levels \(u=\) \(
  +
-3.8,\) \(-3.6,\) \(-3.4,\) \(-3.2\) \(
−
0.2,0.4,0.6,0.8
  
  +
-2.8,\) \(-2.6,\) \(-2.4,\) \(-2.2\) \(
−
1.2,1.4,1.6,1.8
  
−
2.2,2.4,1.6,2.8
 +
-1.8,\) \(-1.6,\) \(-1.4,\) \(-1.2\) \(
−
3.2,3.4,3.6,3.8</math> are shown with thin blue curves.<br>
 +
-0.8,\) \(-0.6,\) \(-0.4,\) \(-0.2\) are shown with thin red curves.<br>
  +
Levels \(u=\) \(
−
Levels <math>v=-24,-20,-16,-12,-8,-7,-6,-5,-4,-3,-2,-1</math> are shown with thick red curves.<br>
  
  +
0.2,\) \(0.4,\) \(0.6,\) \(0.8\) \(
−
Level <math>v=0</math> is shown with thick pink line.<br>
  
  +
1.2,\) \(1.4,\) \(1.6,\) \(1.8\) \(
−
Levels <math>v=1,2,3,4,5,6,7,8,12,16,20,24</math> are shown with thick blue lines.<br>
  
  +
2.2,\) \(2.4,\) \(1.6,\) \(2.8\) \(
−
Levels <math>v=
  
  +
3.2,\) \(3.4,\) \(3.6,\) \(3.8\) are shown with thin blue curves.<br>
−
-3.8,-3.6,-3.4,-3.2
  
  +
Levels \(v=-24,\) \(-20,\) \(-16,\) \(-12,\) \(-8,\) \(-7,\) \(-6,\) \(-5,\) \(-4,\) \(-3,\) \(-2,\) \(-1\) are shown with thick red curves.<br>
−
-2.8,-2.6,-2.4,-2.2
  
  +
Level \(v=0\) is shown with thick pink line.<br>
−
-1.8,-1.6,-1.4,-1.2
  
  +
Levels \(v=1,2,3,4,5,6,7,8,12,16,20,24\) are shown with thick blue lines.<br>
−
-0.8,-0.6,-0.4,-0.2
  
  +
Levels \(v=
−
0.2,0.4,0.6,0.8
  
  +
-3.8,\) \(-3.6,\) \(-3.4,\) \(-3.2\) \(
−
1.2,1.4,1.6,1.8
  
−
2.2,2.4,1.6,2.8
 +
-2.8,\) \(-2.6,\) \(-2.4,\) \(-2.2\) \(
  +
-1.8,\) \(-1.6,\) \(-1.4,\) \(-1.2\) \(
−
3.2,3.4,3.6,3.8</math> are shown with thin green curves.<br>
  
  +
-0.8,\) \(-0.6,\) \(-0.4,\) \(-0.2\) \(
  +
0.2,\) \(0.4,\) \(0.6,\) \(0.8\) \(
  +
1.2,\) \(1.4,\) \(1.6,\) \(1.8\) \(
  +
2.2,\) \(2.4,\) \(1.6,\) \(2.8\) \(
  +
3.2,\) \(3.4,\) \(3.6,\) \(3.8\) are shown with thin green curves.<br>
 
The cut of range of [[holomorphism]] is shown with black dashed line.
  
The cut of range of [[holomorphism]] is shown with black dashed line.
   
−
Function LogFactorial has singularities at the same points, as the factorial, id est, at negative integer values of the argument.
+
Function LogFactorial has singularities at the same points, as the factorial, id est, at negative integer values of the argument.
   
  +
<!--
−
====Approximation of LogFactorial at large values of the argument====
  
  +
===Approximation of LogFactorial at large values of the argument===
  +
!-->
 
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<table border="2" cellpadding="4" cellspacing="2" style="margin: 1em 1em 1em 0; background: #ffffff; border: 1px #aaa solid; border-collapse: collapse;">
  
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<tr> <th><math>n</math></th> <th><math>a_n</math></th> <th>approximation of <math>a_n</math></th> </tr>
 +
<tr> <th>\(n\)</th> <th>\(a_n\)</th> <th>approximation of \(a_n\)</th> </tr>
 
<tr> <td>0</td> <td> 1 / 12 </td> <td>0.083333333333333333</td> </tr>
  
<tr> <td>0</td> <td> 1 / 12 </td> <td>0.083333333333333333</td> </tr>
 
<tr> <td>1</td> <td> 1 / 30 </td> <td>0.033333333333333333</td> </tr>
  
<tr> <td>1</td> <td> 1 / 30 </td> <td>0.033333333333333333</td> </tr>
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</table></div>
  
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Far from the negative part of the real axis, the function LogFactorial can be approximated through the coninual fraction
 
Far from the negative part of the real axis, the function LogFactorial can be approximated through the coninual fraction
  +
\(p\):
−
<math>p</math>:
  
  +
\[
−
: <math>\mathrm{LogFactorial}(z)= p(z) + \log(2\pi)/2 - z + (z+1/2)~\log(z)</math>
  
  +
\mathrm{LogFactorial}(z)= p(z) + \log(2\pi)/2 - z + (z+1/2)~\log(z)
−
: <math>p(z)= \frac{a_0}{z+\frac{a_1}{z+\frac{a_2}{z+\frac{a_3}{z+..}}}}</math>
  
  +
\]
−
The coefficients <math>a</math> and their approximate evaluations are copypasted in the table at right.
  
  +
\[
  +
p(z)= \frac{a_0}{z+\frac{a_1}{z+\frac{a_2}{z+\frac{a_3}{z+..}}}}
  +
\]
  +
The coefficients \(a\) and their approximate evaluations are copypasted in the table at right.
   
   
−
In vicinity of the real axis, while the modulus of the imaginary part of LogFactorial does not exceed <math>\pi</math>, the
 +
In vicinity of the real axis, while the modulus of the imaginary part of LogFactorial does not exceed \(\pi\), the
 
LogFactorial can be interpreted as lofarithm of factorial, id est,
  
LogFactorial can be interpreted as lofarithm of factorial, id est,
−
: <math>\mathrm{LogFactorial}(z)=\log(z!)</math>
 +
: \(\mathrm{LogFactorial}(z)=\log(z!)\)
−
In particular, this relation is valid for positive real values of <math>z</math>.
 +
In particular, this relation is valid for positive real values of \(z\).
   
−
For all <math>z</math> except negative integers, <math> z!=\exp\!\Big(\mathrm{LogFactorial}(z)\Big)</math>
 +
For all \(z\) except negative integers, \( z!=\exp\!\Big(\mathrm{LogFactorial}(z)\Big)\)
   
 
However, the LogFactorial has singularities at negative integer valies of the argument.
  
However, the LogFactorial has singularities at negative integer valies of the argument.
   
−
===Stirling formula===
 +
==[[Stirling formula]]==
  +
 
Historically, one of the first approximations of the factorial with elementary functions was the [[Stirling formula]] below.
  
Historically, one of the first approximations of the factorial with elementary functions was the [[Stirling formula]] below.
−
For large ''n'' there is an approximation due to Scottish mathematician [[James Stirling]]
 +
For large \(n\) there is an [[asymptotic]] approximation due to Scottish mathematician [[James Stirling]]
  +
\[
−
:<math> n! \approx \sqrt{2\pi} n^{n+1/2} e^{-n} . \,</math>
  
  +
n! \approx \sqrt{2\pi} n^{n+1/2} \mathrm e^{-n}
−
This formula can be obtained from the approximation for LogFactorial above, just replacing <math>p(z)</math> to zero.
  
  +
\]
  +
This formula can be obtained from the approximation for LogFactorial above, just replacing \(p(z)\) to zero.
   
  +
With more terms, the [[asymptotic]] \(A\) can be written as flows:
−
==Iterated factorial and square root of factorial==
  
  +
\[
−
Iterated factorial can be constructeed with the
  
  +
A(z)=
−
[[superfunction]] of factiorial and the inverse function, id est, the [[SuperFactorial]] $F$ and the [[AbelFactorial]] $G$. The $c$th iteration of factorial can be expressed as follows:
 
  +
\sqrt{2\pi z}\ \exp\left(
−
: $\mathrm{Factorial}^c(z)= F\big(c+G(z)\big)$
  
  +
z\log\frac{z}{\mathrm e} +
  +
\frac{1}{12 z}
  +
\left(
  +
1
  +
-\frac{1}{30z^2}
  +
+\frac{1}{105z^4}
  +
-\frac{1}{140z^6}
  +
+\frac{1}{99z^8}
  +
-\frac{691}{30030 z^{10}}<!--
  +
+ O(z^{-12})!-->
  +
\right)
  +
\right)
  +
\]
  +
<div class="thumb tright" style="float:right; margin:-8px 0px 8px 2px; background-color:#fff">
  +
{{pic|FactoriAsymptoAgreeT.png|270px}}<small><center>Map of agreement \(a(z\!+\!\mathrm i y)\) in the \(x,y\) plane</center></small>
  +
</div>
  +
The figure at right shows the map of agreement
  +
\[
  +
a(z) = - \lg \left(\frac{|z!-A(z)|}{|z!|+|A(z)|}\right).
  +
\]
  +
This agreement estimates, how may significant figures can one expect to get with this approximation.<br>
  +
The map or the agreement \(a\) suggests that the approximation \(A(z)\)
  +
provides of order of 14 decimal digits of [[Factorial]]\((z)\) in the range
  +
\[ d = (\ \Re(z) \!\ge\! -6 , \ |z|\!>\!8\ )\ \lor \ (\Re(z)\!<\!-6 , \ |\Im(z)|\!>\!5 \ )
  +
\]
  +
  +
Function \(A\) is interpreted as [[Restricted asymptotic]]:<br>
  +
It is the [[Sectorial asymptotic]] of [[Factorial]] at large values for sector
  +
\[
  +
D= \{ z\in \mathbf C \ : |\arg(z)|< \alpha \}
  +
\]
  +
where \(\alpha\) is any positive real number such that \( \alpha < \pi \).
  +
  +
At large \(|z|\),
  +
the residual \(\ r(z)\!=\)[[Factorial]]\((z)\!-\!A(z) \ \)
  +
remains asymptotically negligible compared to [[Factorial]]\((z)\)
  +
in the whole complex \(z\) plane
  +
except an arbitrarily narrow sector containing the negative real axis:
  +
\[
  +
\lim_{|z|\to \infty, \ z \in D} \frac{r(z)}{z!} = 0
  +
\]
  +
Here, the limit is not uniform as \(\alpha\to\pi\) .
  +
The mathematically precise version of the formula above can be written as follows:
  +
\[
  +
\forall \alpha<\pi:\quad
  +
\lim_{|z|\to\infty,\ |\arg z|\le \alpha}
  +
\frac{r(z)}{z!}=0
  +
\]
  +
  +
In a little bit less efficient form, the asymptotic of [[Factorial]] for large values can be written as follows <ref>
  +
https://dlmf.nist.gov/5.11
  +
5 Gamma Function
  +
Properties
  +
As z→∞ in the sector |ph⁡z|≤π−δ, ..
  +
</ref>:
  +
\[
  +
\mathrm{Factorial}(z+1) \approx
  +
B(z) = \sqrt{2\pi z} \, \left(\frac{z}{\mathrm e}\right)^z \left(1 +\frac{1}{12z}+\frac{1}{288 z^2} - \frac{139}{51840 z^3} - \frac{571}{2488320 z^4} + \frac{163879}{209018880 z^5}
  +
+\frac{5246819}{75246796800 z^6}.
  +
\right)
  +
\]
  +
Function \(B\) also can be interpreted as [[Sectorial asymptotic]] of [[Factorial]], with the same sector \(D\) as for asymptotic \(A\).
  +
  +
==Iterated factorial and [[square root of factorial]]==
  +
<div class="thumb tright" style="float:right; margin:-8px 0px 2px 2px; background-color:#fff">
  +
&nbsp; &nbsp; {{pic|FacitT.jpg|300px}}<small><center>\(y=\mathrm{Factorial}^n(x)\) versus \(x\) for various \(n\)</center></small>
  +
</div>
  +
  +
[[Factorial]] is treated as the [[Transfer function]] in book «[[Superfunctions]]»
  +
<ref>
  +
https://www.amazon.co.jp/-/en/Dmitrii-Kouznetsov/dp/6202672862 <br>
  +
https://www.morebooks.de/shop-ui/shop/product/978-620-2-67286-3 <br>
  +
https://mizugadro.mydns.jp/BOOK/468.pdf
  +
D.Kouznetsov. [[Superfunctions]]. [[Lambert Academic Publishing]], 2020.
  +
</ref><ref>
  +
https://mizugadro.mydns.jp/BOOK/202.pdf
  +
Д.Кузнецов. [[Суперфункции]]. [[Lambert Academic Publishing]], 2014.
  +
</ref>. This formalism allows to iterate [[Factorial]] and, in particular, to evaluate
  +
function «[[Square root of factorial]]» \(\varphi\) such that \(\varphi(\varphi(x))=x!\).
  +
  +
The iterates of [[factorial]] are constructed with the
  +
[[superfunction]] of factiorial and the inverse function, id est, the [[SuperFactorial]] \(F\) and the [[AbelFactorial]] \(G=F^{-1}\). The \(n\)th iteration of factorial can be expressed as follows:
  +
  +
\(\mathrm{Factorial}^n(z)= \mathrm{SuperFactorial} \big(n+ \mathrm{AbelFactorial}(z)\big)\)
  +
  +
Where [[SuperFactorial]] is [[superfunction]] of factorial, id est, solution \(F\) of the transfer equation
  +
  +
\( F(z\!+\!1)= \mathrm{Factorial}(F(z))\)
  +
  +
and \(\mathrm{AbelFactorial}=\mathrm{SuperFactorial}^{-1}\), id est, the inverse function. [[AbelFactorial]] is also solution \(G\) of the [[Abel equation]]
  +
  +
\(G(z)+1= G(\mathrm{Factorial}(z))\)
  +
  +
The \(n\)th iteration can be calculated also for non-integer values of \(n\), number of iterations can be complex <ref name="qf">
  +
http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1 D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.</ref>.
  +
  +
In particular, at \(n=1/2\), we have
  +
\({\rm factorial}^{1/2}(z)=\sqrt{\rm factorial}(z)\), where function \(\sqrt{\,!\,}=\sqrt{\rm factorial}\) can be interpreted as [[square root of factorial]]:
  +
\(\sqrt{\,!\,}\Big(\sqrt{\,!\,}(z)\Big)=z!\). Name of this function is used as logo of the [[Physics department of the Moscow State University]] and part of logo of [[Main Page|TORI]] in 2009-2014, until the beginning of the [[Russian invasion into Ukraine]]. (Since 2014, all things related to Russia become toxic; so, the logo had been replaced to formulas for the [[Square root of exponential]].)
  +
  +
Notation \(\sqrt{!} (z)\) happened to be not so convenient. The colleagues, who do not know about Logo of the [[Physics Department of the Moscow State University]], confuse
  +
\(~\sqrt{!} (z)=\sqrt{\mathrm{Factorial}}(z)~\) with \(~\sqrt{\mathrm{Factorial}(z)}=\sqrt{z!}~\).
  +
For this reason, in [[TORI]], notation \(\mathrm{Factorial}^{1/2}(z)\) is used instead.
  +
  +
==Transfer equation==
  +
Factorial can be considered as holomorphic solution \(F\) of the generalized transfer equation
  +
  +
\(F(z\!+\!1)= (z\!+\!1) \, F(z)\)
   
  +
However, \(F=\mathrm{Factorial}\) is not only solution of this equation. Name [[Hadmard finction]] is suggested for one of the other solutions
−
In particular, such an expression at $c=1/2$ determines the
 
  +
<ref>
−
[[Square root of factorial]], id est, half-iteration of factorial;
 
  +
http://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunction.html
−
$\sqrt{!\,}(z)=\mathrm{Factorial}^{1/2}(z)$.
  
  +
Is the Gamma function mis-defined?
−
The square root of factorial is used as logo of the Physics department of the Moscow State University and element of logo of [[TORI]].
  
  +
Or: Hadamard versus Euler — Who found the better Gamma function?
  +
Cited for state for November 2013:
  +
<i>The Gamma function of Euler often is thought as the only function which interpolates the factorial numbers
  +
n!=1,2,6,24,…// This is far from being true. We will discuss four factorial functions//
  +
the Euler factorial function n!,//
  +
the Hadamard Gamma function H(n),//
  +
the logarithmic single inflected factorial function L(n),//
  +
the logarithmic single inflected factorial function L∗(n). ..
  +
</i>
  +
</ref>.
   
 
==References==
  
==References==
  +
{{ref}}
−
<references/>
  
  +
  +
http://en.citizendium.org/wiki?title=Factorial
   
 
<!--* {{cite book | author=Ronald L. Graham | coauthors=Donald E. Knuth, Oren Patashnik | title=Concrete Mathematics | publisher=[[Addison Wesley]] | year=1989 | isbn=0-201-14236-8 | pages=111,332 }}
  
<!--* {{cite book | author=Ronald L. Graham | coauthors=Donald E. Knuth, Oren Patashnik | title=Concrete Mathematics | publisher=[[Addison Wesley]] | year=1989 | isbn=0-201-14236-8 | pages=111,332 }}
 
!-->
  
!-->
−
* Ronald L. Graham, Donald E. Knuth, Oren Patashnik. Concrete Mathematics. (Addison Wesley, 1989, isbn=0-201-14236-8, pages111,332
 +
Ronald L. Graham, Donald E. Knuth, Oren Patashnik. Concrete Mathematics. (Addison Wesley, 1989, isbn=0-201-14236-8, pages111,332
   
  +
https://cyberleninka.ru/article/n/superfunktsii-i-koren-iz-faktoriala/viewer<br>
−
* http://en.citizendium.org/wiki?title=Factorial
 
  +
https://mizugadro.mydns.jp/PAPERS/2010superfar.pdf<br>
  +
Кузнецов Д.Ю.Траппманн Г. Суперфункции и корень из факториала.
  +
[[Вестник Московского Университета]], серия 3, 2010 No.1, стр.8-14.
  +
(Текст научной статьи по специальности «Математика»), in Russian
   
  +
https://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1<br>
  +
https://mizugadro.mydns.jp/PAPERS/2010superfae.pdf<br>
  +
D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial.
  +
[[Moscow University Physics Bulletin]], 2010, v.65, No.1, p.6-12.
  +
  +
{{fer}}
  +
  +
==Keywords==
  +
  +
«[[Amos]]»,
  +
«[[ArcFactorial]]»,
  +
«[[Arcfactorial]]»,
  +
«[[Arcsuperfac.cin]]»,
  +
«[[Asymptotic]]»,
  +
«[[AuFac.cin]]»,
  +
«[[AuFac]]»,
  +
«[[Binomial coefficient]]»,
  +
«[[Fac.cin]]»,
  +
«[[Facp.cin]]»,
  +
«[[Factorial]]»,
  +
«[[Lof]]»,
  +
«[[LogFactorial]]»,
  +
«[[Restricted asymptotic]]»,
  +
«[[Sectorial asymptotic]]»,
  +
«[[Student Distribution]]»,
  +
«[[Square root of factorial]]»,
  +
«[[SuperFactorial]]»,
  +
«[[Superfactorial]]»,
  +
«[[SuFac.cin]]»,
  +
«[[SuFac]]»,
  +
«[[Transfer function]]»,
  +
  +
[[Факториал]]
  +
  +
<!--
 
[[Category:Articles from CZ]]
  
[[Category:Articles from CZ]]
  +
!-->
−
[[Category:Articles in English]]
  
  +
[[Category:English]]
 
[[Category:Factorial]]
  
[[Category:Factorial]]
 
[[Category:Mathematics]]
  
[[Category:Mathematics]]
 
[[Category:Mathematical functions]]
  
[[Category:Mathematical functions]]
  +
[[Category:Transfer function]]
 
[[Category:TORI]]
  
[[Category:TORI]]

Revision as of 20:37, 19 January 2026


\(n\) \(n!\)
\(-5/2\) \(4\sqrt{\pi}/3\)
\(-2\) \(\infty\)
\(-3/2\) -2\(\sqrt{\pi}\)
\(-1\) \(\infty\)
\(-1/2\) \(\sqrt{\pi}\)
0 1
1/2 \(\sqrt{\pi}/2\)
1 1
3/2 \(3 \sqrt{\pi}/4\)
2 2
5/2 \(15 \sqrt{\pi}/8\)
3 6
4 24
5 120
6 720
7 5040
8 40320
9 362880
10 3628800

Warning: This article is under construciton. It still has mistakes.

In mathematics, Factorial is meromorphic function with fast growth along the real axis; for non-negative integer values of the argument, this function has integer values. [1]

Frequently, the postfix notation \(n!\) is used for the factorial of number \(n\). For integer \(n\), the \(n!\) gives the number of ways in which n labelled objects (for example the numbers from 1 to n) can be arranged in order. These are the permutations of the set of objects.

In some programming languages, both n! and factorial(n) , or Factorial(n), are recognized as the factorial of the number \(n\).

Integer values of the argument

For integer values of the argument, the factorial can be defined by a recurrence relation. If n labelled objects have to be assigned to n places, then the n-th object can be placed in one of n places: the remaining n-1 objects then have to be placed in the remaining n-1 places, and this is the same problem for the smaller set. So we have

\( n! = n \cdot (n-1)! \,\)

and it follows that

\( n! = n \cdot (n-1) \cdots 2 \cdot 1 , \,\)

which we could derive directly by noting that the first element can be placed in n ways, the second in n-1 ways, and so on until the last element can be placed in only one remaining way.

Since zero objects can be arranged in just one way ("do nothing") it is conventional to put 0! = 1.

The factorial function is found in many combinatorial counting problems. For example, the binomial coefficients, which count the number of subsets size r drawn from a set of n objects, can be expressed as

\(\binom{n}{r} = \frac{n!}{r! (n-r)!} .\)

The factorial function can be extended to arguments other than positive integers: this gives rise to the Gamma function.

Definitions

For complex values of the argument, the combinatoric definition above should be extended.

Implicit definition

The factorial can be defined as unique meromorphic function \(F\), satisfying relations

\( F(z+1)=(z+1) F(z) \)
\( F(0)=1 \)

for all complex \(z\) except negative integer values. The uniqueness of function \(F\), satisfying these equations, follows from the Wielandt's theorem [2][3]. Historically, the deduction refers to the Gamma function, but the application to the factorial is straightforward.


Definition through the integral

Usually, the integral representation is used as definition. For \(\Re(z)>-1\), define

\( z! = \int_0^\infty t^z \exp(-t) \mathrm{d}t \)

Such definition is similar to that of the Gamma function, and leads to the relation

\(z!=\Gamma(z+1)\)

for all complex \(z\) except the negative integer values.

The definition above agrees with the combinatoric definition for integer values of the argument; at integer \(z\), the integral can be expressed in terms of the elementary functions.

Extension of integral definition

Factorialz.jpg
\(f=z!\) in the complex \(z\)-plane.

The definition through the integral can be extended to the whole complex plane, using relation

\(z!=(z+1)!/(z+1)\)

for the cases \(\Re(z)<-1\), assuming that \(z\) is not negative integer. Also, the symmetry formula takes place for the non-integer values of \(z\),

\(z! (-z)! =\frac{\pi z}{\sin(\pi z)}\)

The similar formula of symmetry holds however, for the Gamma function. From this expression, it follows that \(\frac{1}{z!(-z)!}\) is entire function of \(z\). Also, this symmetry gives the simple way to express \((1/2)!=\sqrt{\pi}/2\), and, therefore, factorial of half-integer numbers.

In the figure, lines of constant \(u=\Re(z!)\) and lines of constant \(v=\Im(z!)\) are shown.
The levels \(u = − 24,\) \(− 20,\) \(− 16,\) \(− 12,\) \(− 8,\) \(− 7,\) \(− 6, \) \(− 5,\) \(− 4,\) \(− 3,\) \(− 2,\) \(− 1,\) \( 0,\) \(1,\) \(2,\) \(3,\) \(4,\) \(5,\) \(6,\) \(7,\) \(8,\) \(12,\) \(16,\) \(20,\) \(24\) are drown with thick black lines.
Some of intermediate levels u = const are shown with thin blue lines for positive values and with thin red lines for negative values.
Levels \(v = − 24,\) \(− 20,\) \(− 16,\) \(− 12,\) \(− 8,\) \( − 7,\) \( − 6,\) \( − 5,\) \( − 4,\) \( − 3,\) \( − 2,\) \( − 1\) are shown with thick red lines.
Level \(v = 0\) is shown with thick pink lines.
Levels \(v = 1,\) \( 2,\) \( 3,\) \( 4,\) \( 5,\) \( 6,\) \( 7,\) \( 8,\) \( 12,\) \( 16,\) \( 20,\) \( 24~\) are drown with thick blue lines. some of intermediate levels \(v = \mathrm{const}\) are shown with thin green lines.
The dashed blue line shows the level \(u=\mu_0\) and corresponds to the value \(\mu_0=(x_0)!\approx 0.85\) of the principal local minimum \(x_0\approx 0.45\) of the factorial of the real argument.
The dashed red line shows the level and corresponds to the similar value of the negative local extremum of the factorial of the real argument.
Due to the fast growth of the function, in the right hand side of the figure, the density of the levels exceeds the ability of the plotter to draw them; so, this part is left empty.

Factorial of the real argument

FactoReal.jpg
\(\mathrm{factorial}(x)=x!\); the inverse function, i.e., \(\mathrm{factorial}^{-1}(x)\), and \(\mathrm{factorial}(x)^{-1}\) versus real \(x\).

The definition above was elaborated for factorial of complex argument. In particular, it can be used to evlauate the factorial of the real argument. In the figure at right, the \(\mathrm{factorial}(x)=x!\) is plotted versus real \(x\) with red line. The function has simple poluses at negative integer \(x\).

At \(x\rightarrow -1+o\), the \(x! \rightarrow +\infty\).

The local minimum

The factorial has local minimum at

\(x=\nu_0\approx 0.4616321450\)

marked in the picture with pink vertical line; at this point, the derivative of the factorial is zero:

\(\mathrm{factorial}^{\prime}(\nu_0)=0\)

The value of factorial in this point

\(\mu_0=\nu_0!=\mathrm{factorial(\nu_0)}\approx 0.8856031944\)

The Tailor expansion of \(z!\) at the point \(z=\nu_0\) can be writen ax follows:

\(z!=\mu_0+\sum_{n=2}^{N-1} c_n (z-\nu_0)^n + \mathcal{O}(z-\nu_0)^N~\) .

The coefficients of this expansion are copypasted in the table below:

\(n\) approximation of \(c_n\)
2 0.428486815855585429730209907810650582960483696962
3 -0.130704158939785761928008749242671025181542078103
4 0.160890753325112844190519489594363387594505844657
5 -0.092277030213334350126864106458600575084335085690

This expansion can be used for the precise evaluation of the inverse function of factorial (arcfactorial) in vicinity of the branchpoint.

Other local extremums are at negative values of the argument; one of them in shown in the figure above. There, it is denoted as \(\nu_1\).


The Taylor expansion

The Taylor expansion of \(z!\) at \(z=0\), or the MacLaurin expansion, has the form \(z!=\sum_{n=0}^{N-1} g_n z^n+\mathcal{O}(z^N)~\) . The first coefficients of this expansion are copypasted in the table below:

\(n\) \(g_n\) approximation of \(g_n\)
0 \(1\) \(\ 1.000000000000000000000000000000000000000000000000000000\)
1\(~-\gamma~\) \(-0.5772156649015328606065120900824024310421593359399235988\)
2 \(\frac{\pi^2}{12}+\frac{\gamma^2}{2}\) \(\ 0.9890559953279725553953956515006347079391835207282140904\)
3\(-\frac{\zeta(3)}{3}-\frac{\pi^2\gamma}{12}+\frac{\gamma^3}{6}\) \(-0.9074790760808862890165601673562751149286114490725637609\)
4\(\frac{\pi^4}{160}+\frac{\zeta(3)}{3}-\frac{\pi^2\gamma^2}{12}+\frac{\gamma^4}{24}\) \(\ 0.9817280868344001873363802940218508503605736797234654154\)

Here, \(\gamma\) is the Euler constant and \(\zeta\) is the Riemann function. The Computer algebra systems such as Maple and Mathematica can generate many terms of this expansion. The radius of convergence of the Taylor series is unity, and the coefficient \(g_n\) does not decay as \(n\) increases. However, due to the relation \(z!=z\cdot \mathrm{factorial}(z)\), for any real value of argument of factorial, the expansion above can be used for the precize evaluation of factorial of the real argument, running the approximation above for an argument with modulus not larget than halh. Also, the expansion at the half-integer values can be used: \(z!=\sum_{n=0}^{N-1} h_n~\left(z-\frac{1}{2}\right)^n+\mathcal{O}\left(\left(z-\frac{1}{2}\right)^N\right)~\) . The first coefficients of this expansion are copypasted in the table below:

\(n\) \(h_n\) approximation of \(g_n\)
0 \(\frac{\sqrt{\pi}}{2}\) \( 0.88622692545275801364908374167057259139877472806 \)
1\(\frac{-\sqrt{\pi}}{2}(\gamma+\log(4)-2)\) \( 0.03233839744888501382886988426897030778133478887\)
2 \(\frac{\pi^{5/2}}{8}+\sqrt{\pi}\gamma -\sqrt{\pi}\log(4) +\frac{\gamma^2\sqrt{\pi}}{4} +\sqrt{\pi}\log(2)^2\) \( 0.41481345368830116823003762311135634284890996337\)
3 long expression \(-0.10729480456477221168754195638970966205457592382\)
4even longer expression \( 0.14464535904462154303833221025388452407002686153\)

The Taylor series of \(z!\) developed at \(z=1/2\), converges for all \(z\) such that \(|z-1/2|<3/2\).

In order to boost the approximation of factorial for real values of the argument, and, especially, for the evaluation for complex values, and the evaluation of the inverse function, the expansions for \(\log(z!)\) and \(1/z!\) are used instead of the direct Taylor expansions above.

Halphing of the imaginary part of the argument

While neither \(z\) nor \((z+1)/2\) is negative integer, the argument of factorial can be dropped with the identity

\( \mathrm{Factorial}(z)~=~\frac{2^z}{\sqrt{\pi}}~ \mathrm{Factorial}\!\left(\frac{z}{2} \right)\cdot \mathrm{Factorial}\!\left(\frac{z\!-\!1}{2} \right) \)

This may be useful to drop with factor 1/2 the imaginary part of the argument (for example, to apply the Taylor expansion above for the evaluation), but the function has to be evaluated twice.

Related functions

ArcGamma.jpg
ArcFactorial in the complex plane.

In the plot of factorial of the real argument, the two other functions are plotted, \(\mathrm{ArcFactorial}(x)=\mathrm{factorial}^{-1}(x)\) and \(\mathrm{factorial}(x)^{-1}\). These functions are used for the generalization of the factorial and for its evaluation.

Inverse function

\(z\) ArcFactorial\((z)\)
\(\mu_0\) 0.46163214496836234126265954233
\(\frac{\sqrt{\pi}}{2}\) 0.50000000000000000000000000000
1 1.00000000000000000000000000000
2 2.00000000000000000000000000000
3 2.40586998630956692469992921838
4 2.66403279720644615568638939436
5 2.85235545803172783164299808684
6 3.00000000000000000000000000000

Inverse function of factorial can be defined with equation

\((\mathrm{ArcFactorial}(z))!=z\)

and condition that ArcFactorial is holomorphic in the comlex plane with cut along the part of the real axis, that begins at the minimum value of the factorial of the positive argument, and extends to \(-\infty\). This function is shown with lines of constant real part \(u=\Re(\mathrm{ArcFactorial}(z))\) and lines of constant imaginary part \(v=\Im(\mathrm{ArcFactorial}(z))\).

Levels \(u=1,2,3\) are shown with thick black curves.
Levels \(u= 0.2,\) \(0.4,\) \(0.6,\) \(0.8,\) \( 1.2,\) \(1.4,\) \(1.6,\) \(1.8,\) \( 2.2,\) \(2.4,\) \(2.6,\) \(2.8, 3.2,\) \(3.4,\) \(3.6\) are shown with thin blue curves.
Levels \(v=1,2,3\) are shown with thick blue curves.
Level \(v=0\) is shown with thick pink line.
Levels \(v=-1,-2,-3\) are shown with thick red curves.
The intermediate levels of constant \(v\) are shown with thin dark green curves. The ArcFactorial has the branch point \(\mu_0 \approx 0.85 \); the cut of the range of holomorphizm is shown with black dashed line.

ArcFactorial for some real values of the argument is approximated in the table at right.

\(n\) approximation for \(d_n\)
1 1.43764234228440800
2 0.315227181071631549
3 -0.0256407066268564423
4 0.00492170392390056555

In vicinity of the branchpoint \(z=\mu_0\), the ArcFactorial can be expanded as follows:

\( \mathrm{ArcFactorial}(z)=\nu_0+\sum_{n=1}^{N-1} d_n\cdot \Big(\log(z/\mu_0) \Big)^{n/2} \)

The approximations for the first coefficients of this expansion are copypasted in the table at right. About of 30 terms in this expansion are sufficient to plot the distribution of the real and the imaginary parts of ArcFactorial in the figure.



Function f(z)=1/z!

OneOverFactorial.jpg
\(f(z)=\frac{1}{z!}~\) in the complex \(z\)-plane.

The inverse function of factorial, id est, \(\mathrm{ArcFactorial}(z)=\mathrm{Factorial}^{-1}(z)~\) from the previous section, sohuld not be confused with

\(f(z)=\) \(\frac{1}{z!}\) \( =\mathrm{Factorial}(z)^{-1}\) \(=\frac{1}{\mathrm{Factorial}(z)}~\), which is also shown in the figure at right. The lines of constant \(u=\Re(f(z))\) and the lines of constant \(v=\Im(f(z))\) are drawn.
The levels \(u=-24,\) \(-20,\) \(-16,\) \(-12,\) \(-8,\) \(-7\) \( .. 7,\) \(8,\) \(12,\) \(16,\) \(20,\) \(24\) are shown with thick black lines.
The levels \(v=-24,-20,-16,-12,-8,-7 ... 7,-1\) are shown with thick red lines.
The level \(v=0\) is shown with thick pink line.
The levels \(v=1,2, ... 7,8,12,16,20,24\) are shown with thick blue lines.
Some of intermediate elvels \(u=\)const are shown with thin red lines for negative values and thin blue lines for the positive values.
Some of intermediate elvels \(v=\)const are shown with thin green lines.
The blue dashed curves represent the level \(u=1/\mu_0\) and correspond to the positive local maximum of the inverse function of the real argument.
The red dashed curves represent the level \(u=1/\mu_1\), which corresponds to the first negative local maximum of the factorial of the real argument; \(\mathrm{factorial}^{\prime}(\nu_1)=0\); \(\mathrm{factorial}(\nu_1)=\mu_1\).

In the upper-left hand side of the figure, and at the lower-left hand side of the figure, the density of levels exceeds the ability of the ploter to draw them, and these parts are left empty.

\(f(z)=\frac{1}{z!}\) is entire function, that grows in the left hand side of the compelx plane and quickly decays to zero along the real axis.


LogFactorial

LogFactorialZ.jpg
LogFactorial at the complex plane

For the approximation of factorial, it can be expressed as \[ z!=\exp(\mathrm{LogFactorial}(z)) \] At large values of the argument, the LogFactorial does not show the fast growth and is easier for the numerical implementation, than factorial.

At real positive values of the argument, LogFactorial can be interpreted as logarithm of Factorial: \[ \mathrm{LogFactorial}(x)=\ln(\mathrm{LogFactorial}(x)), \ x>0 \]

Function LogFactorial is shown in figure a right with lines of constant real part and lines of constant imaginary part. Levels of constant \(u=\Re(\mathrm{LogFactorial}(z))\) and Levels of constant \(v=\Im(\mathrm{LogFactorial}(z))\) are drawn with solid lines:
Levels \(u=-24,\) \(-20,\) \(-16,\) \(-12,\) \(-8,\) \(-7,\) \(-6,\) \(-5,\) \(-4,\) \(-3,\) \(-2,\) \(-1,\) \(0,\) \(1,\) \(2,\) \(3,\) \(4,\) \(5,\) \(6,\) \(7,\) \(8,\) \(12\) \(,16,\) \(20,\) \(24\) are shown with thick black curves.
Levels \(u=\) \( -3.8,\) \(-3.6,\) \(-3.4,\) \(-3.2\) \( -2.8,\) \(-2.6,\) \(-2.4,\) \(-2.2\) \( -1.8,\) \(-1.6,\) \(-1.4,\) \(-1.2\) \( -0.8,\) \(-0.6,\) \(-0.4,\) \(-0.2\) are shown with thin red curves.
Levels \(u=\) \( 0.2,\) \(0.4,\) \(0.6,\) \(0.8\) \( 1.2,\) \(1.4,\) \(1.6,\) \(1.8\) \( 2.2,\) \(2.4,\) \(1.6,\) \(2.8\) \( 3.2,\) \(3.4,\) \(3.6,\) \(3.8\) are shown with thin blue curves.
Levels \(v=-24,\) \(-20,\) \(-16,\) \(-12,\) \(-8,\) \(-7,\) \(-6,\) \(-5,\) \(-4,\) \(-3,\) \(-2,\) \(-1\) are shown with thick red curves.
Level \(v=0\) is shown with thick pink line.
Levels \(v=1,2,3,4,5,6,7,8,12,16,20,24\) are shown with thick blue lines.
Levels \(v= -3.8,\) \(-3.6,\) \(-3.4,\) \(-3.2\) \( -2.8,\) \(-2.6,\) \(-2.4,\) \(-2.2\) \( -1.8,\) \(-1.6,\) \(-1.4,\) \(-1.2\) \( -0.8,\) \(-0.6,\) \(-0.4,\) \(-0.2\) \( 0.2,\) \(0.4,\) \(0.6,\) \(0.8\) \( 1.2,\) \(1.4,\) \(1.6,\) \(1.8\) \( 2.2,\) \(2.4,\) \(1.6,\) \(2.8\) \( 3.2,\) \(3.4,\) \(3.6,\) \(3.8\) are shown with thin green curves.
The cut of range of holomorphism is shown with black dashed line.

Function LogFactorial has singularities at the same points, as the factorial, id est, at negative integer values of the argument.

\(n\) \(a_n\) approximation of \(a_n\)
0 1 / 12 0.083333333333333333
1 1 / 30 0.033333333333333333
2 53 / 210 0.252380952380952381
3 195 / 371 0.525606469002695418

Far from the negative part of the real axis, the function LogFactorial can be approximated through the coninual fraction \(p\): \[ \mathrm{LogFactorial}(z)= p(z) + \log(2\pi)/2 - z + (z+1/2)~\log(z) \] \[ p(z)= \frac{a_0}{z+\frac{a_1}{z+\frac{a_2}{z+\frac{a_3}{z+..}}}} \] The coefficients \(a\) and their approximate evaluations are copypasted in the table at right.


In vicinity of the real axis, while the modulus of the imaginary part of LogFactorial does not exceed \(\pi\), the LogFactorial can be interpreted as lofarithm of factorial, id est,

\(\mathrm{LogFactorial}(z)=\log(z!)\)

In particular, this relation is valid for positive real values of \(z\).

For all \(z\) except negative integers, \( z!=\exp\!\Big(\mathrm{LogFactorial}(z)\Big)\)

However, the LogFactorial has singularities at negative integer valies of the argument.

Stirling formula

Historically, one of the first approximations of the factorial with elementary functions was the Stirling formula below. For large \(n\) there is an asymptotic approximation due to Scottish mathematician James Stirling \[ n! \approx \sqrt{2\pi} n^{n+1/2} \mathrm e^{-n} \] This formula can be obtained from the approximation for LogFactorial above, just replacing \(p(z)\) to zero.

With more terms, the asymptotic \(A\) can be written as flows: \[ A(z)= \sqrt{2\pi z}\ \exp\left( z\log\frac{z}{\mathrm e} + \frac{1}{12 z} \left( 1 -\frac{1}{30z^2} +\frac{1}{105z^4} -\frac{1}{140z^6} +\frac{1}{99z^8} -\frac{691}{30030 z^{10}} \right) \right) \]

FactoriAsymptoAgreeT.png
Map of agreement \(a(z\!+\!\mathrm i y)\) in the \(x,y\) plane

The figure at right shows the map of agreement \[ a(z) = - \lg \left(\frac{|z!-A(z)|}{|z!|+|A(z)|}\right). \] This agreement estimates, how may significant figures can one expect to get with this approximation.
The map or the agreement \(a\) suggests that the approximation \(A(z)\) provides of order of 14 decimal digits of Factorial\((z)\) in the range \[ d = (\ \Re(z) \!\ge\! -6 , \ |z|\!>\!8\ )\ \lor \ (\Re(z)\!<\!-6 , \ |\Im(z)|\!>\!5 \ ) \]

Function \(A\) is interpreted as Restricted asymptotic:
It is the Sectorial asymptotic of Factorial at large values for sector \[ D= \{ z\in \mathbf C \ : |\arg(z)|< \alpha \} \] where \(\alpha\) is any positive real number such that \( \alpha < \pi \).

At large \(|z|\), the residual \(\ r(z)\!=\)Factorial\((z)\!-\!A(z) \ \) remains asymptotically negligible compared to Factorial\((z)\) in the whole complex \(z\) plane except an arbitrarily narrow sector containing the negative real axis: \[ \lim_{|z|\to \infty, \ z \in D} \frac{r(z)}{z!} = 0 \] Here, the limit is not uniform as \(\alpha\to\pi\) . The mathematically precise version of the formula above can be written as follows: \[ \forall \alpha<\pi:\quad \lim_{|z|\to\infty,\ |\arg z|\le \alpha} \frac{r(z)}{z!}=0 \]

In a little bit less efficient form, the asymptotic of Factorial for large values can be written as follows [4]: \[ \mathrm{Factorial}(z+1) \approx B(z) = \sqrt{2\pi z} \, \left(\frac{z}{\mathrm e}\right)^z \left(1 +\frac{1}{12z}+\frac{1}{288 z^2} - \frac{139}{51840 z^3} - \frac{571}{2488320 z^4} + \frac{163879}{209018880 z^5} +\frac{5246819}{75246796800 z^6}. \right) \] Function \(B\) also can be interpreted as Sectorial asymptotic of Factorial, with the same sector \(D\) as for asymptotic \(A\).

Iterated factorial and square root of factorial

    FacitT.jpg
\(y=\mathrm{Factorial}^n(x)\) versus \(x\) for various \(n\)

Factorial is treated as the Transfer function in book «Superfunctions» [5][6]. This formalism allows to iterate Factorial and, in particular, to evaluate function «Square root of factorial» \(\varphi\) such that \(\varphi(\varphi(x))=x!\).

The iterates of factorial are constructed with the superfunction of factiorial and the inverse function, id est, the SuperFactorial \(F\) and the AbelFactorial \(G=F^{-1}\). The \(n\)th iteration of factorial can be expressed as follows:

\(\mathrm{Factorial}^n(z)= \mathrm{SuperFactorial} \big(n+ \mathrm{AbelFactorial}(z)\big)\)

Where SuperFactorial is superfunction of factorial, id est, solution \(F\) of the transfer equation

\( F(z\!+\!1)= \mathrm{Factorial}(F(z))\)

and \(\mathrm{AbelFactorial}=\mathrm{SuperFactorial}^{-1}\), id est, the inverse function. AbelFactorial is also solution \(G\) of the Abel equation

\(G(z)+1= G(\mathrm{Factorial}(z))\)

The \(n\)th iteration can be calculated also for non-integer values of \(n\), number of iterations can be complex [7].

In particular, at \(n=1/2\), we have \({\rm factorial}^{1/2}(z)=\sqrt{\rm factorial}(z)\), where function \(\sqrt{\,!\,}=\sqrt{\rm factorial}\) can be interpreted as square root of factorial: \(\sqrt{\,!\,}\Big(\sqrt{\,!\,}(z)\Big)=z!\). Name of this function is used as logo of the Physics department of the Moscow State University and part of logo of TORI in 2009-2014, until the beginning of the Russian invasion into Ukraine. (Since 2014, all things related to Russia become toxic; so, the logo had been replaced to formulas for the Square root of exponential.)

Notation \(\sqrt{!} (z)\) happened to be not so convenient. The colleagues, who do not know about Logo of the Physics Department of the Moscow State University, confuse \(~\sqrt{!} (z)=\sqrt{\mathrm{Factorial}}(z)~\) with \(~\sqrt{\mathrm{Factorial}(z)}=\sqrt{z!}~\). For this reason, in TORI, notation \(\mathrm{Factorial}^{1/2}(z)\) is used instead.

Transfer equation

Factorial can be considered as holomorphic solution \(F\) of the generalized transfer equation

\(F(z\!+\!1)= (z\!+\!1) \, F(z)\)

However, \(F=\mathrm{Factorial}\) is not only solution of this equation. Name Hadmard finction is suggested for one of the other solutions [8].

References

  1. ↑ http://people.math.sfu.ca/~cbm/aands/page_255.htm M.Abramowitz and I.Stegun: Handbook of Mathematical Functions. 6. Gamma Function and Related Functions (2010)
  2. ↑ Wielandt's theorem online: http://www.math.ku.dk/~henrikp/specialtopic2000/node7.html
  3. ↑ Reinhold Remmert The American Mathematical Monthly, Vol. 103, No. 3 (Mar., 1996), pp. 214-220, http://www.jstor.org/pss/2975370
  4. ↑ https://dlmf.nist.gov/5.11 5 Gamma Function Properties As z→∞ in the sector |ph⁡z|≤π−δ, ..
  5. ↑ https://www.amazon.co.jp/-/en/Dmitrii-Kouznetsov/dp/6202672862
    https://www.morebooks.de/shop-ui/shop/product/978-620-2-67286-3
    https://mizugadro.mydns.jp/BOOK/468.pdf D.Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.
  6. ↑ https://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014.
  7. ↑ http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1 D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.
  8. ↑ http://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunction.html Is the Gamma function mis-defined? Or: Hadamard versus Euler — Who found the better Gamma function? Cited for state for November 2013: The Gamma function of Euler often is thought as the only function which interpolates the factorial numbers n!=1,2,6,24,…// This is far from being true. We will discuss four factorial functions// the Euler factorial function n!,// the Hadamard Gamma function H(n),// the logarithmic single inflected factorial function L(n),// the logarithmic single inflected factorial function L∗(n). ..

http://en.citizendium.org/wiki?title=Factorial

Ronald L. Graham, Donald E. Knuth, Oren Patashnik. Concrete Mathematics. (Addison Wesley, 1989, isbn=0-201-14236-8, pages111,332

https://cyberleninka.ru/article/n/superfunktsii-i-koren-iz-faktoriala/viewer
https://mizugadro.mydns.jp/PAPERS/2010superfar.pdf
Кузнецов Д.Ю.Траппманн Г. Суперфункции и корень из факториала. Вестник Московского Университета, серия 3, 2010 No.1, стр.8-14. (Текст научной статьи по специальности «Математика»), in Russian

https://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1
https://mizugadro.mydns.jp/PAPERS/2010superfae.pdf
D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.

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