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8 Incomplete Gamma and Related FunctionsIncomplete Gamma Functions

§8.11 Asymptotic Approximations and Expansions

  1. §8.11(i) Large z, Fixed a
  2. §8.11(ii) Large a, Fixed z
  3. §8.11(iii) Large a, Fixed z/a
  4. §8.11(iv) Large a, Bounded (xa)/(2a)12
  5. §8.11(v) Other Approximations

§8.11(i) Large z, Fixed a


8.11.1 uk=(1)k(1a)k=(a1)(a2)(ak),
8.11.2 Γ(a,z)=za1ez(k=0n1ukzk+Rn(a,z)),

Then as z with a and n fixed

8.11.3 Rn(a,z)=O(zn),

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 na1. 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)=zaezk=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|πδ.


8.11.5 P(a,z)zaezΓ(1+a)(2πa)12eaz(z/a)a,
a, |pha|πδ.

§8.11(iii) Large a, Fixed z/a

If z=λa, with λ fixed, then as a

8.11.6 γ(a,z)zaezk=0(a)kbk(λ)(za)2k+1,
0<λ<1, |pha|π2δ.
8.11.7 Γ(a,z)zaezk=0(a)kbk(λ)(za)2k+1,
λ>1, |pha|3π2δ.


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(1λ)bk1(λ)+(2k1)λbk1(λ).

Sharp error bounds and an exponentially-improved extension for (8.11.7) can be found in Nemes (2016). This reference also contains explicit formulas for bk(λ) in terms of Stirling numbers and for the case λ>1 an asymptotic expansion for bk(λ) as k.

The expansion (8.11.7) also applies when a is replaced by a, λ<0 and |pha|3π2ωδ with ω=ph(λ+ln(λ)+πi), 0<ω<π. For error bounds and an exponentially-improved extension for this later expansion, see Nemes (2015c). In the case that a=n, a positive integer, the z-region of validity of (8.11.7) is discussed in Ameur and Cronvall (2023).

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

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

8.11.10 P(a+1,x)=12erfc(y)132πa(1+y2)ey2+O(a1),
8.11.11 γ(1a,x)=xa1(cos(πa)+sin(πa)π(2πF(y)+232πa(1y2))ey2+O(a1)),

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)zz1ez(π2z1213+2π24z124135z+2π576z32+82835z2+),

For sharp error bounds and an exponentially-improved extension, see Nemes (2016). This reference also contains explicit formulas for the coefficients in terms of Stirling numbers.

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)nenx.

As n

8.11.16 Sn(1)12n!ennn23+4135n182835n2168505n3+,
8.11.17 Sn(1)12+18n1+132n21128n313512n4+.


8.11.18 Sn(x)k=0dk(x)nk,

uniformly for x(,1δ], with

8.11.19 d0(x) =x/(1x),
dk(x) =(1)kbk(x)(1x)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).