Stirling formula

Stirling formula

Reprinted from Baidu Encyclopedia    

https://baike.baidu.com/item/Sterling formula

Stirling's approximation is a mathematical formula used to approximate the factorial of n. Generally speaking, when n is large, the calculation of n factorial is very large, so Stirling's formula is very easy to use, and even when n is small, the value of Stirling's formula is very accurate.

definition

Stirling's approximation is a mathematical formula used to approximate the factorial of n. Generally speaking, when n is large, the calculation of n factorial is very large, so Stirling's formula is very easy to use, and even when n is small, the value of Stirling's formula is very accurate.

Stirling's formula has important value in theory and application, and also has great significance for the development of probability theory . In mathematical analysis , most of them use the knowledge of Г function, series and integral with parameter variables to prove or deduce, which is very cumbersome and lengthy. In recent years, some scholars at home and abroad have used the exponential distribution , Poisson distribution and χ² distribution in probability theory to prove it.

form

or more precise

or

Prove

make

 

but

 

so

   

which is

   

, that is, monotonically decreasing, and by the integral scaling method, we have

 

which is

   

,Right now

 

By the monotone bound theorem

   

The limit exists [1]    ,

Assume

 

Use Wallis Official ,

 

so

 

which is