Thursday, October 1, 2015

asymptotics - Growth of Gamma(n,n)



How can you get the asymptotics for the growth of Γ(n,n)?



Γ(n,n)=nxn1exp(x)dx


Answer



You can get it by a simple rescaling of the integral. Let x=nt; then



Γ(n,n)=nn1dttn1ent=nn1dtten(logtt)=nnen0du1+uen[log(1+u)u]



The contribution to this integral is dominated by that near u=0 as n. We may then Taylor expand the term in the exponential and see that the leading asymptotic behavior of the integral is



Γ(n,n)nnen0duenu2/2=nnenπ2n




We may find further terms in the asymptotic behavior by Taylor expanding the higher-order terms in the exponential and the term outside the exponential. The next higher order term is



(1+nu33)(1u)1u+nu33



Evaluating the integrals that result from this expansion, we find the next higher term in the expansion:



Γ(n,n)=nnen[π2n13n+O(n3/2)]


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