Stirling's approximation
Stirling's formula

Under these conditions, we may use Stirling's approximation for the factorial:

These functions often contain terms with factorials which scale as (Stirling's approximation).

(This can be proved by using Stirling's approximation for n.)

With the help of Stirling's approximation, entropy can be simplified:

From Stirling's approximation we know that is .

Substituting Stirling's approximation in this expression (both for k!

Using Stirling's approximation of the factorial one can establish the following asymptotic result:

After applying Stirling's approximation for the factorial of a large integer m:

A more precise description of the high dimensional behavior of the volume can be obtained using Stirling's approximation.

Stirling's approximation is a well-known asymptotic formula for the factorial function:

For large n we get a better estimate for the number n using Stirling's approximation:

(The assumption that would be invoked in such calculation, which allows one to apply Stirling's approximation.)

If approximate values of factorial numbers are desired, Stirling's approximation gives good results using floating-point arithmetic.

Call this randomly selected permutation G. Note from Stirling's approximation that (2)!

Stirling's approximation for n!

Stirling's approximation is in fact an underestimate of the gamma function, so the above formula is an upper bound.

It is a practical alternative to the more popular Stirling's approximation for calculating the Gamma function with fixed precision.

You can improve on Stirling's approximation systematically by using the Euler-Mclaurin series for conversion of a sum to an integral.

The Stirling numbers, Stirling permutations, and Stirling's approximation are named after him.

Second, it is convenient to maximize the natural log (ln) rather than w(Tij), for then we may use Stirling's approximation.

Therefore, Stirling's approximation is as much due to de Moivre as it is to Stirling.

In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for large factorials.

Including more terms from the zeta series yields a better approximation, as does factoring in the asymptotic series in Stirling's approximation.

That expansion, in turn, serves as the starting point for one of the derivations of precise error estimates for Stirling's approximation of the factorial function.

The entropy of mixing is also proportional to the Shannon entropy or compositional uncertainty of information theory, which is defined without requiring Stirling's approximation.