Evaluate numerically:

Evaluate to high precision:

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

Complex number input:

Evaluate efficiently at high precision:

Subfactorial automatically threads over lists:

Compute the elementwise values of an array using automatic threading:

Or compute the matrix Subfactorial function using MatrixFunction:

Subfactorial can be used with CenteredInterval objects:

Values of Subfactorial at fixed points:

Value at zero:

Evaluate symbolically:

Limiting values at infinity:

Find a value of for which the real part of Subfactorial[x] is equal to 5:

Plot the absolute value of Subfactorial:

Plot the real part of Subfactorial[z]:

Plot the imaginary part of Subfactorial[z]:

Real domain of Subfactorial:

Complex domain:

Subfactorial is not an analytic function on :

In fact, it is singular and discontinuous everywhere on the reals:

The reason is that it is only real valued at isolated points:

However, it is analytic in the complex plane:

The imaginary part of Subfactorial is not injective:

The imaginary part of Subfactorial is not surjective:

The real part of Subfactorial is neither non-negative nor non-positive:

The real part of Subfactorial is neither convex nor concave:

First derivative with respect to :

Higher derivatives with respect to :

Plot the higher derivatives with respect to :

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Find the series expansion at Infinity:

Taylor expansion at a generic point:

One-step recurrence relation:

Two-step recurrence relation:

On the positive integers, Subfactorial[n]==Round[n!/E]: