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8 Incomplete Gamma and Related FunctionsRelated Functions

§8.20 Asymptotic Expansions of Ep(z)


§8.20(i) Large z

8.20.1 Ep(z)=e-zz(k=0n-1(-1)k(p)kzk+(-1)n(p)nezzn-1En+p(z)),

As z

8.20.2 Ep(z)e-zzk=0(-1)k(p)kzk,


8.20.3 Ep(z)±2πiΓ(p)epπizp-1+e-zzk=0(-1)k(p)kzk,

δ again denoting an arbitrary small positive constant. Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii).

For an exponentially-improved asymptotic expansion of Ep(z) see §2.11(iii).

§8.20(ii) Large p

For x0 and p>1 let x=λp and define A0(λ)=1,

8.20.4 Ak+1(λ)=(1-2kλ)Ak(λ)+λ(λ+1)dAk(λ)dλ,

so that Ak(λ) is a polynomial in λ of degree k-1 when k1. In particular,

8.20.5 A1(λ) =1,
A2(λ) =1-2λ,
A3(λ) =1-8λ+6λ2.

Then as p

8.20.6 Ep(λp)e-λp(λ+1)pk=0Ak(λ)(λ+1)2k1pk,

uniformly for λ[0,).

For further information, including extensions to complex values of x and p, see Temme (1994b, §4) and Dunster (1996b, 1997).