Factorial

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. The uniqueness of function $F$ , satisfying these equations, follows from the Wielandt's theorem ^[1] ^[2]. Historialy, 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

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!=\Gamma(z+1)}

for all 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} except the negative integer values.

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

Extension of integral definition

f=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 definition through the integral can be extended to the whole complex plane, using 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+1)!/(z+1)}

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

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)! =\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 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} . Also, this simmetry gives the simple way to express 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 (1/2)!=\sqrt{\pi}/2} , and, therefore, factorial of half-integer numbers.

In the figure, lines of constant $u=\Re (z!)$ and 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(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 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=\mu_0} and corresponds to the value 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=(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

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)=x!} ; the inverse function, 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{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}} 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 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)=x!} is plotted versus real 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} with red line. The function has simple poluses at negative integer 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} .

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:

\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 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=\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:

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 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} 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 $g_{n}$
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}}	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 0.989055995327972555395395651500634707939183520728214090443192}
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

(CC) Image: Dmitrii Kouznetsov
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 the factorial 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.

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 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 d_n}
1	1.43764234228440800
2	0.315227181071631549
3	-0.0256407066268564423
4	0.00492170392390056555

In vicinity of the branchpoint 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} , the ArcFactorial can be expanded as 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 \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 polt the distribution of the real and the imaginary parts of ArcFactorial in the figure above.

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!}

(CC) Image: Dmitrii Kouznetsov
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} .

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.

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

Template:CC-Image
LogFactorial at the complex plane.

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)) }

Plof of LogFactorial

Function LogFactorial is shown in figure with lines of constant real part and lines of constant imaginary part. 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 u=\Re(\mathrm{LogFactorial}(z))} and 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=\Im(\mathrm{LogFactorial}(z))} are drawn with solid lines:
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,-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 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= -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 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 2.2,2.4,1.6,2.8 3.2,3.4,3.6,3.8} 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=-24,-20,-16,-12,-8,-7,-6,-5,-4,-3,-2,-1} are shown with thick red 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=1,2,3,4,5,6,7,8,12,16,20,24} are shown with thick blue lines.
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= -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.

Approximation of LogFactorial at large values of the 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 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

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 at right.

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} .

For all $z$ except negative integers, 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\!\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 approximation due to Scottish mathematician James Stirling

n!\approx {\sqrt {2\pi }}n^{n+1/2}e^{-n}.\,

This formula can be obtained from the approximation for LogFactorial 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.

↑ Wielandt's theorem online: http://www.math.ku.dk/~henrikp/specialtopic2000/node7.html
↑ Reinhold Remmert The American Mathematical Monthly, Vol. 103, No. 3 (Mar., 1996), pp. 214-220, http://www.jstor.org/pss/2975370

[Wielandt's_theorem-1] Wielandt's theorem online: http://www.math.ku.dk/~henrikp/specialtopic2000/node7.html

[reinhold-2] Reinhold Remmert The American Mathematical Monthly, Vol. 103, No. 3 (Mar., 1996), pp. 214-220, http://www.jstor.org/pss/2975370

[1]

[2]

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

Factorial

Contents

Integer values of the argument