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8 Incomplete Gamma and Related FunctionsRelated Functions

§8.18 Asymptotic Expansions of Ix(a,b)


§8.18(i) Large Parameters, Fixed x

If b and x are fixed, with b>0 and 0<x<1, then as a

8.18.1 Ix(a,b)=Γ(a+b)xa(1-x)b-1(k=0n-11Γ(a+k+1)Γ(b-k)(x1-x)k+O(1Γ(a+n+1))),

for each n=0,1,2,. If b=1,2,3, and nb, then the O-term can be omitted and the result is exact.

If b and a and x are fixed, with a>0 and 0<x<1, then (8.18.1), with a and b interchanged and x replaced by 1-x, can be combined with (8.17.4).

§8.18(ii) Large Parameters: Uniform Asymptotic Expansions

Large a, Fixed b


8.18.2 ξ=-lnx.

Then as a, with b (>0) fixed,

8.18.3 Ix(a,b)=Γ(a+b)Γ(a)(k=0n-1dkFk+O(a-n)F0),

uniformly for x(0,1]. The functions Fk are defined by

8.18.4 aFk+1=(k+b-aξ)Fk+kξFk-1,


8.18.5 F0 =a-bQ(b,aξ),
F1 =b-aξaF0+ξbe-aξaΓ(b),

and Q(a,z) as in §8.2(i). The coefficients dk are defined by the generating function

8.18.6 (1-e-tt)b-1=k=0dk(t-ξ)k.

In particular,

8.18.7 d0 =(1-xξ)b-1,
d1 =xξ+x-1(1-x)ξ(b-1)d0.

Compare also §24.16(i). A recurrence relation for the dk can be found in Nemes and Olde Daalhuis (2016).

Symmetric Case


8.18.8 x0=a/(a+b).

Then as a+b,

8.18.9 Ix(a,b)12erfc(-ηb/2)+12π(a+b)(xx0)a(1-x1-x0)bk=0(-1)kck(η)(a+b)k,

uniformly for x(0,1) and a/(a+b), b/(a+b)[δ,1-δ], where δ again denotes an arbitrary small positive constant. For erfc see §7.2(i). Also,

8.18.10 -12η2=x0ln(xx0)+(1-x0)ln(1-x1-x0),

with η/(x-x0)>0, and

8.18.11 c0(η)=1η-x0(1-x0)x-x0,

with limiting value

8.18.12 c0(0)=1-2x03x0(1-x0).

For this result, and for higher coefficients ck(η) see Temme (1996b, § All of the ck(η) are analytic at η=0.

General Case

Let Γ~(z) denote the scaled gamma function

8.18.13 Γ~(z)=(2π)-1/2ezz(1/2)-zΓ(z),

μ=b/a, and x0 again be as in (8.18.8). Then as a

8.18.14 Ix(a,b)Q(b,aζ)-(2πb)-1/2Γ~(b)(xx0)a(1-x1-x0)bk=0hk(ζ,μ)ak,

uniformly for b(0,) and x(0,1). Here

8.18.15 μlnζ-ζ=lnx+μln(1-x)+(1+μ)ln(1+μ)-μ,

with (ζ-μ)/(x0-x)>0, and

8.18.16 h0(ζ,μ)=μ(1ζ-μ-(1+μ)-3/2x0-x),

with limiting value

8.18.17 h0(μ,μ)=13(1-μ1+μ-1).

For this result and higher coefficients hk(ζ,μ) see Temme (1996b, § All of the hk(ζ,μ) are analytic at ζ=μ (corresponding to x=x0).

Inverse Function

For asymptotic expansions for large values of a and/or b of the x-solution of the equation

8.18.18 Ix(a,b)=p,

see Temme (1992b).