Factorial

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Specific values

Specialized values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

n! is an analytical function of n which is defined in the whole complex n-plane with the exception of countably many points n=-k/;k∈(SuperPlus[ℕ]). 1/n! is an entire function.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

The function n! has an infinite set of singular points:
a) n==-k/;k∈(SuperPlus[ℕ]) are the simple poles with residues ((((-1))^(k-1))/((k-1)!));
b) n==(OverTilde[∞]) is the point of convergence of poles, which is an essential singular point.

Branch points

The function n! does not have branch points.

Branch cuts

The function n! does not have branch cuts.

Series representations

Generalized power series

Expansions at n==(Subscript[n, 0])/; (Subscript[n, 0])≠-m

Expansions at n==-m

Asymptotic series expansions

Product representations

Limit representations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Multiple arguments

Products, sums, and powers of the direct function

Products of the direct function

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations of special kind

n! is the unique nonzero solution of the functional equation f(n)==n f(n-1) which is logarithmically convex for all real n>0; that is, for which log(f(n)) is a convex function for n>0.

Differentiation

Low-order differentiation

Symbolic differentiation

Fractional integro-differentiation

Summation

Finite summation

H. J. Brothers

Infinite summation

Parameter-free sums

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Parameter-containing sums

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Operations

Limit operation

Representations through more general functions

Through other functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With related functions

Inequalities

Zeros

Theorems

Taylor's formula

f(z)==(∑_(k=0)^∞)((Derivative[k][f](a) (((z-a))^k))/(k!)).

Derivative of composition (Faà di Bruno's formula)

(((∂^n)f(g(x)))/(∂(x^n)))==(∑_(m=1)^n)(∑_(Underscript[(Subscript[k, 1])+ (Subscript[k, 2])+…+ (Subscript[k, n])=m, (Subscript[k, 1])+ 2(Subscript[k, 2])+…+ n (Subscript[k, n])=n])^)Derivative[m][f](g(x)) ((n!)/((∏_(j=1)^n)((j!)^(Subscript[k, j])) (Subscript[k, j])!)) (∏_(j=1)^n)((Derivative[j][g](x))^(Subscript[k, j]))==(∑_(m=1)^n)((1 )/(m!))((∑_(j=0)^(m-1))(((-1))^j) (({{m}, {j}})) ((g(x))^j) (((∂^n)((g(x))^(m-j)))/(∂(x^n))))Derivative[m][f](g(x)).

Composition of two series

(∑_(m=0)^∞)(((Subscript[b, m])((∑_(k=1)^∞)(Subscript[a, k])(z^k)))^m)==(∑_(n=0)^∞)(Subscript[c, n])(z^n)/;(Subscript[c, 0])==(Subscript[b, 0])∧(Subscript[c, 1])==(Subscript[a, 1])(Subscript[b, 1])∧(Subscript[c, 2])==(Subscript[a, 1]^2)(Subscript[b, 2])+(Subscript[a, 2])(Subscript[b, 1])∧(Subscript[c, 3])==(Subscript[a, 3])(Subscript[b, 1])+2(Subscript[a, 1])(Subscript[a, 2])(Subscript[b, 2])+(Subscript[a, 1]^3)(Subscript[b, 3])∧(Subscript[c, 4])==(Subscript[a, 4])(Subscript[b, 1])+2(Subscript[a, 1])(Subscript[a, 3])(Subscript[b, 2])+(Subscript[a, 2]^2)(Subscript[b, 2])+3(Subscript[a, 1]^2)(Subscript[a, 2])(Subscript[b, 3])+(Subscript[a, 1]^4)(Subscript[b, 4])∧(Subscript[c, n])==(∑_(Underscript[(Subscript[k, 1]),(Subscript[k, 2]),…,(Subscript[k, n])=0, (Subscript[k, 1])+ 2(Subscript[k, 2])+…+ n (Subscript[k, n])=n])^(n))m! (Subscript[b, m])(∏_(j=1)^n)((Subscript[a, j]^(Subscript[k, j]))/((Subscript[k, j])!)).∧m==(∑_(j=1)^n)(Subscript[k, j])

Maxfield theorem and Castell conjecture

J. E. Maxfield proved that the base 10 digits of any positive integer occur in m! as the first digits for some integer m (J. E. Maxfield. Math. Mag. 43, 64, (1970)). 
Castell's conjecture states that the digits 1 to b-1 of the base b expansion of n! are asymptotically equally distributed (S. P. Castell. Eureka, n36, 44 1973).

History

– J. Stirling (1730) found his famous asymptotic formula
– L. Euler (1751)
– C. Kramp (1808, 1816) introduced the notation n!

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