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8 Incomplete Gamma and Related FunctionsIncomplete Gamma Functions

§8.11 Asymptotic Approximations and Expansions


§8.11(i) Large z, Fixed a


8.11.1 uk=(-1)k(1-a)k=(a-1)(a-2)(a-k),
8.11.2 Γ(a,z)=za-1e-z(k=0n-1ukzk+Rn(a,z)),

Then as z with a and n fixed

8.11.3 Rn(a,z)=O(z-n),

where δ denotes an arbitrary small positive constant.

If a is real and z (=x) is positive, then Rn(a,x) is bounded in absolute value by the first neglected term un/xn and has the same sign provided that na-1. For bounds on Rn(a,z) when a is real and z is complex see Olver (1997b, pp. 109–112). For an exponentially-improved asymptotic expansion (§2.11(iii)) see Olver (1991a).

§8.11(ii) Large a, Fixed z

8.11.4 γ(a,z)=zae-zk=0zk(a)k+1,

This expansion is absolutely convergent for all finite z, and it can also be regarded as a generalized asymptotic expansion (§2.1(v)) of γ(a,z) as a in |pha|π-δ.


8.11.5 P(a,z)zae-zΓ(1+a)(2πa)-12ea-z(z/a)a,
a, |pha|π-δ.

§8.11(iii) Large a, Fixed x/a

If x=λa, with λ fixed, then as a+

8.11.6 γ(a,x)-xae-xk=0(-a)kbk(λ)(x-a)2k+1,
8.11.7 Γ(a,x)xae-xk=0(-a)kbk(λ)(x-a)2k+1,


8.11.8 b0(λ) =1,
b1(λ) =λ,
b2(λ) =λ(2λ+1),

and for k=1,2,,

8.11.9 bk(λ)=λ(1-λ)bk-1(λ)+(2k-1)λbk-1(λ).

The expansion (8.11.7) also applies when a- with λ<0, and in this case Gautschi (1959a) supplies numerical bounds for the remainders in the truncated expansion (8.11.7). For extensions to complex variables see Temme (1994b, §4), and also Mahler (1930), Tricomi (1950b), and Paris (2002b).

§8.11(iv) Large a, Bounded (x-a)/(2a)12

If x=a+(2a)12y and a+, then

8.11.10 P(a+1,x)=12erfc(-y)-132πa(1+y2)e-y2+O(a-1),
8.11.11 γ*(1-a,-x)=xa-1(-cos(πa)+sin(πa)π(2πF(y)+232πa(1-y2))ey2+O(a-1)),

in both cases uniformly with respect to bounded real values of y. For Dawson’s integral F(y) see §7.2(ii). See Tricomi (1950b) for these approximations, together with higher terms and extensions to complex variables. For related expansions involving Hermite polynomials see Pagurova (1965).

§8.11(v) Other Approximations

As z,

8.11.12 Γ(z,z)zz-1e-z(π2z12-13+2π24z12-4135z+2π576z32+82835z2+),

For the function en(z) defined by (8.4.11),

8.11.13 limnen(nx)enx={0,x>1,12,x=1,1,0x<1.

With x=1, an asymptotic expansion of en(nx)/enx follows from (8.11.14) and (8.11.16).

If Sn(x) is defined by

8.11.14 enx=en(nx)+(nx)nn!Sn(x),


8.11.15 Sn(x)=γ(n+1,nx)(nx)ne-nx.

As n

8.11.16 Sn(1)-12n!ennn-23+4135n-1-82835n-2-168505n-3+,
8.11.17 Sn(-1)-12+18n-1+132n-2-1128n-3-13512n-4+.


8.11.18 Sn(x)k=0dk(x)n-k,

uniformly for x(-,1-δ], with

8.11.19 dk(x)=(-1)kbk(x)(1-x)2k+1,

and bk(x) as in §8.11(iii).

For (8.11.18) and extensions to complex values of x see Buckholtz (1963). For a uniformly valid expansion for n and x[δ,1], see Wong (1973b).