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8 Incomplete Gamma and Related FunctionsIncomplete Gamma Functions

§8.12 Uniform Asymptotic Expansions for Large Parameter

Define

8.12.1 λ =z/a,
η =(2(λ-1-lnλ))1/2,

where the branch of the square root is continuous and satisfies η(λ)λ-1 as λ1. Then

8.12.2 12η2 =λ-1-lnλ,
ηλ =λ-1λη.

Also, denote

8.12.3 P(a,z)=12erfc(-ηa/2)-S(a,η),
8.12.4 Q(a,z)=12erfc(ηa/2)+S(a,η),
8.12.5 Γ(a+1)±πa2πΓ(-a,z±π)=12erfc(±ηa/2)+T(a,η),

and

8.12.6 z-aγ*(-a,-z)=cos(πa)-2sin(πa)(12aη2πF(ηa/2)+T(a,η)),

where F(x) is Dawson’s integral; see §7.2(ii). Then as a in the sector |pha|π-δ(<π),

8.12.7 S(a,η)-12aη22πak=0ck(η)a-k,
8.12.8 T(a,η)12aη22πak=0ck(η)(-a)-k,

in each case uniformly with respect to λ in the sector |phλ|2π-δ (<2π).

With μ=λ-1, the coefficients ck(η) are given by

8.12.9 c0(η) =1μ-1η,
c1(η) =1η3-1μ3-1μ2-112μ,
8.12.10 ck(η)=1ηηck-1(η)+(-1)kgkμ,
k=1,2,,

where gk, k=0,1,2,, are the coefficients that appear in the asymptotic expansion (5.11.3) of Γ(z). The right-hand sides of equations (8.12.9), (8.12.10) have removable singularities at η=0, and the Maclaurin series expansion of ck(η) is given by

8.12.11 ck(η)=n=0dk,nηn,
|η|<2π,

where d0,0=-13,

8.12.12 d0,n =(n+2)αn+2,
n1,
dk,n =(-1)kgkd0,n+(n+2)dk-1,n+2,
n0, k1,

and α3,α4, are defined by

8.12.13 λ-1=η+13η2+n=3αnηn,
|η|<2π.

In particular,

8.12.14 α3 =136,
α4 =-1270,
α5 =14320,
α6 =117010,
α7 =-13954 43200,
α8 =12 04120.

For numerical values of dk,n to 30D for k=0(1)9 and n=0(1)Nk, where Nk=28-4k/2, see DiDonato and Morris (1986).

Special cases are given by

8.12.15 Q(a,a)12+12πak=0ck(0)a-k,
|pha|π-δ,
8.12.16 ±πa2sin(πa)Q(-a,a±π)±12-2πak=0ck(0)(-a)-k,
|pha|π-δ,

where

8.12.17 c0(0) =-13,
c1(0) =-1540,
c2(0) =256048,
c3(0) =1011 55520,
c4(0) =-31 8481136951 55200,
c5(0) =-27 4549381517 36320.

For error bounds for (8.12.7) see Paris (2002a). For the asymptotic behavior of ck(η) as k see Dunster et al. (1998) and Olde Daalhuis (1998c). The last reference also includes an exponentially-improved version (§2.11(iii)) of the expansions (8.12.4) and (8.12.7) for Q(a,z).

A different type of uniform expansion with coefficients that do not possess a removable singularity at z=a is given by

8.12.18 Q(a,z)P(a,z)}za-12-zΓ(a)(d(±χ)k=0Ak(χ)zk/2±k=1Bk(χ)zk/2),

for z in |phz|<12π, with (z-a)0 for P(a,z) and (z-a)0 for Q(a,z). Here

8.12.19 χ =(z-a)/z,
d(±χ) =12πχ2/2erfc(±χ/2),

and

8.12.20 A0(χ) =1,
A1(χ) =12χ+16χ3,
B1(χ) =13+16χ2.

Higher coefficients Ak(χ), Bk(χ), up to k=8, are given in Paris (2002b).

Lastly, a uniform approximation for Γ(a,ax) for large a, with error bounds, can be found in Dunster (1996a).

For other uniform asymptotic approximations of the incomplete gamma functions in terms of the function erfc see Paris (2002b) and Dunster (1996a).

Inverse Function

For asymptotic expansions, as a, of the inverse function x=x(a,q) that satisfies the equation

8.12.21 Q(a,x)=q

see Temme (1992a). These expansions involve the inverse error function inverfc(x)7.17), and are uniform with respect to q[0,1]. As a special case,

8.12.22 x(a,12)a-13+8405a-1+18425515a-2+224834 44525a-3+,
a.