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8 Incomplete Gamma and Related FunctionsRelated Functions

§8.18 Asymptotic Expansions of Ix⁑(a,b)

  1. Β§8.18(i) Large Parameters, Fixed x
  2. Β§8.18(ii) Large Parameters: Uniform Asymptotic Expansions

Β§8.18(i) Large Parameters, Fixed x

If b and x are fixed, with b>0 and 0<x<1, then as aβ†’βˆž

8.18.1 Ix⁑(a,b)=Γ⁑(a+b)⁒xa⁒(1βˆ’x)bβˆ’1⁒(βˆ‘k=0nβˆ’11Γ⁑(a+k+1)⁒Γ⁑(bβˆ’k)⁒(x1βˆ’x)k+O⁑(1Γ⁑(a+n+1))),

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

If bβ†’βˆž 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 1βˆ’x, can be combined with (8.17.4).

Β§8.18(ii) Large Parameters: Uniform Asymptotic Expansions

Large a, Fixed b


8.18.2 ΞΎ=βˆ’ln⁑x.

Then as aβ†’βˆž, with b (>0) fixed,

8.18.3 Ix⁑(a,b)=Γ⁑(a+b)Γ⁑(a)⁒(βˆ‘k=0nβˆ’1dk⁒Fk+O⁑(aβˆ’n)⁒F0),

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

8.18.4 a⁒Fk+1=(k+bβˆ’a⁒ξ)⁒Fk+k⁒ξ⁒Fkβˆ’1,


8.18.5 F0 =aβˆ’b⁒Q⁑(b,a⁒ξ),
F1 =bβˆ’a⁒ξa⁒F0+ΞΎb⁒eβˆ’a⁒ξa⁒Γ⁑(b),

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

8.18.6 (1βˆ’eβˆ’tt)bβˆ’1=βˆ‘k=0∞dk⁒(tβˆ’ΞΎ)k.

In particular,

8.18.7 d0 =(1βˆ’xΞΎ)bβˆ’1,
d1 =x⁒ξ+xβˆ’1(1βˆ’x)⁒ξ⁒(bβˆ’1)⁒d0.

Compare also Β§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+bβ†’βˆž,

8.18.9 Ix⁑(a,b)∼12⁒erfc⁑(βˆ’Ξ·β’(a+b)/2)+12⁒π⁒(a+b)⁒(xx0)a⁒(1βˆ’x1βˆ’x0)bβ’βˆ‘k=0∞(βˆ’1)k⁒ck⁑(Ξ·)(a+b)k,

uniformly for x∈(0,1) and a/(a+b), b/(a+b)∈[Ξ΄,1βˆ’Ξ΄], where Ξ΄ again denotes an arbitrary small positive constant. For erfc see Β§7.2(i). Also,

8.18.10 βˆ’12⁒η2=x0⁒ln⁑(xx0)+(1βˆ’x0)⁒ln⁑(1βˆ’x1βˆ’x0),

with Ξ·/(xβˆ’x0)>0, and

8.18.11 c0⁑(Ξ·)=1Ξ·βˆ’x0⁒(1βˆ’x0)xβˆ’x0,

with limiting value

8.18.12 c0⁑(0)=1βˆ’2⁒x03⁒x0⁒(1βˆ’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

For the scaled gamma function Ξ“βˆ—β‘(z) see (5.11.3).

8.18.13 See (5.11.3).

Let ΞΌ=b/a, and x0 again be as in (8.18.8). Then as aβ†’βˆž

8.18.14 Ix⁑(a,b)∼Q⁑(b,a⁒΢)βˆ’(2⁒π⁒b)βˆ’1/2Ξ“βˆ—β‘(b)⁒(xx0)a⁒(1βˆ’x1βˆ’x0)bβ’βˆ‘k=0∞hk⁑(ΞΆ,ΞΌ)ak,

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

8.18.15 μ⁒lnβ‘ΞΆβˆ’ΞΆ=ln⁑x+μ⁒ln⁑(1βˆ’x)+(1+ΞΌ)⁒ln⁑(1+ΞΌ)βˆ’ΞΌ,

with (ΞΆβˆ’ΞΌ)/(x0βˆ’x)>0, and

8.18.16 h0⁑(ΞΆ,ΞΌ)=μ⁒(1ΞΆβˆ’ΞΌβˆ’(1+ΞΌ)βˆ’3/2x0βˆ’x),

with limiting value

8.18.17 h0⁑(ΞΌ,ΞΌ)=13⁒(1βˆ’ΞΌ1+ΞΌβˆ’1).

For this result and higher coefficients hk⁑(΢,μ) see Temme (1996b, § All of the hk⁑(΢,μ) are analytic at ΢=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).