Factorial

$n$	$n!$
0	1
1	1
2	2
3	6
4	24
5	120
6	720
7	5040
8	40320
9	362880
10	3628800

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. 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 definiton 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. This definition is not constructive, and gives no straightforward way for the evaluation.

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

$f=z!$ 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 $z$ in not negative integer. This allows to plot the map of factorial in the complex plane. 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.
The levels v = − 24, − 20, − 16, − 12, − 8, − 7, − 6, − 5, − 4, − 3, − 2, − 1 are shown with thick red lines.
The level v = 0 is shown with thick pink lines.
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.
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

$\mathrm {factorial} (x)=x!$; the inverse function, id est, $\mathrm {factorial} ^{-1}(x)$, 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x! \rightarrow +\infty} .

The local minimum

The factorial has local minimum at

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\nu_0\approx 0.461632144968362341262659542325721328468196204}

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{factorial}^{\prime}(\nu_0)=0}

The value of factorial in this point

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_0=\nu_0!=\mathrm{factorial(\nu_0)}\approx 0.88560319441088870027881590058258873320795153367}

The Tailor expansion of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z!} at the point $z=\nu _{0}$ can be writen ax follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	approximation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.

The Taylor expansion

The Taylor expansion of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z!} at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	$g_{n}$	approximation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_n}
0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1.000000000000000000000000000000000000000000000000000000000000}
1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\gamma}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -0.577215664901532860606512090082402431042159335939923598805767}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\pi^2}{12}+\frac{\gamma^2}{2}}	$0.989055995327972555395395651500634707939183520728214090443192$
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\zeta(3)}{3}-\frac{\pi^2\gamma}{12}+\frac{\gamma^3}{6}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -0.907479076080886289016560167356275114928611449072563760941328}
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\pi^4}{160}+\frac{\zeta(3)}{3}-\frac{\pi^2\gamma^2}{12}+\frac{\gamma^4}{24}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.981728086834400187336380294021850850360573679723465415404953}

Here, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} is the Euler constant and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_n} does not decay as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} increases. However, due to the relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_n}	approximation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_n}
0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\sqrt{\pi}}{2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.886226925452758013649083741670572591398774728061193564106905 }
1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{-\sqrt{\pi}}{2}(\gamma+\log(4)-2)}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.0323383974488850138288698842689703077813347888705070206366386}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\pi^{5/2}}{8}+\sqrt{\pi}\gamma -\sqrt{\pi}\log(4) +\frac{\gamma^2\sqrt{\pi}}{4} +\sqrt{\pi}\log(2)^2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.414813453688301168230037623111356342848909963370422367977736}
3	long expression	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -0.107294804564772211687541956389709662054575923821298300938631}
4	even longer expression	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.144645359044621543038332210253884524070026861530981428414028}

The Taylor series of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z!} developed at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=1/2} , converges for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z-1/2)<3/2} .

In order to boost the approximation of factorial for real values of the argument, and, expecially, for the evaluation for complex values, and the evaluation of the inverse function, the expansions for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log(z!)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/z!} are used instead of the direct Taylor expansions above.

Related functions

ArcFactorial in the complex plane.

In the plot of factorial of the real argument, the two other functions are plotted, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ArcFactorial}(x)=\mathrm{factorial}^{-1}(x)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{factorial}(x)^{-1}} . These functions can be useful for the generalization of factiroal and for its evaluation.

Inverse function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z}	ArcFactorialFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (z)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_0}	0.46163214496836234126265954233
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\infty} . This function is shown with lines of constant real part Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=\Re(\mathrm{ArcFactorial}(z))} and lines of constant imaginary part Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v=\Im(\mathrm{ArcFactorial}(z))} .

Levels Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=1,2,3} are shown with thick black curves.
Levels Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v=1,2,3} are shown with thick blue curves.
Level Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v=0} is shown with thick pink line.
Levels Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v=-1,-2,-3} are shown with thick red curves.
The intermediate levels of constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} are shown with thin dark green curves. The ArcFactorial has the branch point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.

Function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z)=1/z!}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z)=\frac{1}{z!}} in the complex Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} -plane.

The inverse function of factorial, id est, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ArcFactorial}(z)=\mathrm{Factorial}^{-1}(z)} from the previous section, sohuld not be confused with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z)=\frac{1}{z!}=\mathrm{Factorial}(z)^{-1}=\frac{1}{\mathrm{Factorial}(z)}} shown in the figure at right. The lines of constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=\Re(f(z))} and the lines of constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v=\Im(f(z))} are drawn.
The levels Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=-24,-20,-16,-12,-8,-7 .. 7,8,12,16,20,24} are shown with thick black lines.
The levels Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v=-24,-20,-16,-12,-8,-7 ... 7,-1} are shown with thick red lines.
The level Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v=0} is shown with thick pink line.
The levels Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v=1,2, ... 7,8,12,16,20,24} are shown with thick blue lines.
Some of intermediate elvels Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=} const are shown with thin red lines for negative values and thin blue lines for the positive values.
Some of intermediate elvels Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v=} const are shown with thin green lines.
The blue dashed curves represent the level Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=1/\mu_1} , which corresponds to the first negative local maximum of the factorial of the real argument; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{factorial}^{\prime}(\nu_1)=0} ; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{factorial}(\nu_1)=\mu_1} .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

For the approximation of factorial, if can be represented in the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z!=\exp(\mathrm{LogFactorial}(z)) }

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

Far from the negative part of the real axis, the function LogFactorial can be approximated through the coninual fraction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{LogFactorial}(z)= p(z) + \log(2\pi)/2 - z + (z+1/2)~\log(z)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(z)= \frac{a_0}{z+\frac{a_1}{z+\frac{a_2}{z+\frac{a_3}{z+..}}}}}

The coefficients Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and their approximate evaluations are copypasted in the table below:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n}	approximation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n}
0	1 / 12	0.083333333333333333
1	1 / 30	0.033333333333333333
2	53 / 210	0.252380952380952381
3	195 / 371	0.525606469002695418

In vicinity of the real axis, while the modulus of the imaginary part of LogFactorial does not exceed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} , the LogFactorial can be interpreted as lofarithm of factorial, id est,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{LogFactorial}(z)=\log(z!)}

In particular, this relation is valid for positive real values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} .

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 approximation due to Scottish mathematician James Stirling

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n! \approx \sqrt{2\pi} n^{n+1/2} e^{-n} . \,}

This formula can be obtained from the approximation for LogFactgorial above, just replacing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(z)} to zero.

References

• Ronald L. Graham; Donald E. Knuth, Oren Patashnik (1989). Concrete Mathematics. Addison Wesley, 111,332. ISBN 0-201-14236-8.