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.

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

Factorial can be evaluated to arbitrary numerical precision.

Factorial automatically threads over lists.

Out[1]=

Out[2]=

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

-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: