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单词 StirlingsApproximation
释义

Stirling’s approximation


Stirling’s formula gives an approximation for n!, the factorial . It is

n!2nπnne-n

We can derive this from the gamma functionDlmfDlmfMathworldPlanetmath. Note that for large x,

Γ(x)=2πxx-12e-x+μ(x)(1)

where

μ(x)=n=0(x+n+12)ln(1+1x+n)-1=θ12x

with 0<θ<1. Taking x=n and multiplying by n, we have

n!=2πnn+12e-n+θ12n(2)

Taking the approximation for large n gives us Stirling’s formula.

There is also a big-O notation version of Stirling’s approximation:

n!=(2πn)(ne)n(1+𝒪(1n))(3)

We can prove this equality starting from (2). It is clear that the big-O portion of (3) must come from eθ12n, so we must consider the asymptotic behavior of e.

First we observe that the Taylor seriesMathworldPlanetmath for ex is

ex=1+x1+x22!+x33!+

But in our case we have e to a vanishing exponentMathworldPlanetmathPlanetmath. Note that if we vary x as 1n, we have as n

ex=1+𝒪(1n)

We can then (almost) directly plug this in to (2) to get (3) (note that the factor of 12 gets absorbed by the big-O notation.)

TitleStirling’s approximation
Canonical nameStirlingsApproximation
Date of creation2013-03-22 12:00:36
Last modified on2013-03-22 12:00:36
Ownerdrini (3)
Last modified bydrini (3)
Numerical id22
Authordrini (3)
Entry typeTheorem
Classificationmsc 68Q25
Classificationmsc 30E15
Classificationmsc 41A60
SynonymStirling’s formula
SynonymStirling’s approximation formula
Related topicMinkowskisConstant
Related topicAsymptoticBoundsForFactorial
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