Built-in Mathematica Symbol

Factorial (!)

n!
gives the factorial of n.

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

For non-integer n, the numerical value of n! is given by Gamma[1+n].

For integers and half integers, Factorial automatically evaluates to exact values.

Factorial can be evaluated to arbitrary numerical precision.

Factorial automatically threads over lists.

Scope (7)

Evaluate for large arguments:

Evaluate for half-integer arguments:

Plot the factorial function:

Evaluate numerically for complex arguments:

Factorial threads element-wise over lists:

Series expansion at infinity (Stirling's approximation):

Series at

:

Generalizations & Extensions (4)

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Infinite arguments give symbolic results:

Factorial allows derivatives:

Applications (6)

Make a table of half-integer factorials:

Number of permutations of 6 elements:

Plot of the absolute value of Factorial in the complex plane:

Find the asymptotic expansion of ratios of factorials:

Volume of an n-dimensional unit hypersphere:

Low-dimensional cases:

Plot the volume of the unit hypersphere as a function of dimension:

Find the series expansion at -

:

Properties & Relations (6)

Use FullSimplify to simplify expressions involving Factorial:

Compute a generating function sum involving Factorial:

Compute numerical sums involving Factorial:

The generating function is divergent:

Use regularization to obtain a closed form generating function:

Generating function as a formal series:

Some integrals can be done:

Product of factorials:

Possible Issues (2)

Large arguments can give results too large to be computed explicitly, even approximately:

Smaller values work:

Machine-number inputs can give high-precision results:

Neat Examples (3)

Find the numbers of digits 0 through 9 in 100000!:

Nested factorials over the complex plane:

Plot Factorial at infinity: