What's New
About the Project
NIST
5 Gamma FunctionProperties

§5.11 Asymptotic Expansions

Contents

§5.11(i) Poincaré-Type Expansions

As z in the sector |phz|π-δ(<π),

5.11.1 lnΓ(z)(z-12)lnz-z+12ln(2π)+k=1B2k2k(2k-1)z2k-1

and

5.11.2 ψ(z)lnz-12z-k=1B2k2kz2k.

For the Bernoulli numbers B2k, see §24.2(i).

With the same conditions,

5.11.3 Γ(z)e-zzz(2πz)1/2(k=0gkzk),

where

5.11.4 g0 =1,
g1 =112,
g2 =1288,
g3 =-13951840,
g4 =-57124 88320,
g5 =1 638792090 18880,
g6 =52 468197 52467 96800.

Also,

5.11.5 gk=2(12)ka2k,

where a0=122 and

5.11.6 a0ak+12a1ak-1+13a2ak-2++1k+1aka0=1kak-1,
k1.

Wrench (1968) gives exact values of gk up to g20. Spira (1971) corrects errors in Wrench’s results and also supplies exact and 45D values of gk for k=21,22,,30. For an asymptotic expansion of gk as k see Boyd (1994).

Terminology

The expansion (5.11.1) is called Stirling’s series (Whittaker and Watson (1927, §12.33)), whereas the expansion (5.11.3), or sometimes just its leading term, is known as Stirling’s formula (Abramowitz and Stegun (1964, §6.1), Olver (1997b, p. 88)).

Next, and again with the same conditions,

5.11.7 Γ(az+b)2πe-az(az)az+b-(1/2),

where a(>0) and b() are both fixed, and

5.11.8 lnΓ(z+h)(z+h-12)lnz-z+12ln(2π)+k=2(-1)kBk(h)k(k-1)zk-1,

where h([0,1]) is fixed, and Bk(h) is the Bernoulli polynomial defined in §24.2(i).

Lastly, as y±,

5.11.9 |Γ(x+iy)|2π|y|x-(1/2)e-π|y|/2,

uniformly for bounded real values of x.

§5.11(ii) Error Bounds and Exponential Improvement

If the sums in the expansions (5.11.1) and (5.11.2) are terminated at k=n-1 (k0) and z is real and positive, then the remainder terms are bounded in magnitude by the first neglected terms and have the same sign. If z is complex, then the remainder terms are bounded in magnitude by sec2n(12phz) for (5.11.1), and sec2n+1(12phz) for (5.11.2), times the first neglected terms.

For the remainder term in (5.11.3) write

5.11.10 Γ(z)=e-zzz(2πz)1/2(k=0K-1gkzk+RK(z)),
K=1,2,3,.

Then

5.11.11 |RK(z)|(1+ζ(K))Γ(K)2(2π)K+1|z|K(1+min(sec(phz),2K12)),
|phz|12π,

where ζ(K) is as in Chapter 25. In the case K=1 the factor 1+ζ(K) is replaced with 4. For this result and a similar bound for the sector 12πphzπ see Boyd (1994).

For further information see Olver (1997b, pp. 293–295), and for other error bounds see Whittaker and Watson (1927, §12.33), Spira (1971), and Schäfke and Finsterer (1990).

For re-expansions of the remainder terms in (5.11.1) and (5.11.3) in series of incomplete gamma functions with exponential improvement (§2.11(iii)) in the asymptotic expansions, see Berry (1991), Boyd (1994), and Paris and Kaminski (2001, §6.4).

§5.11(iii) Ratios

In this subsection a, b, and c are real or complex constants.

If z in the sector |phz|π-δ (<π), then

5.11.12 Γ(z+a)Γ(z+b) za-b,
5.11.13 Γ(z+a)Γ(z+b) za-bk=0Gk(a,b)zk.

Also, with the added condition (b-a)>0,

5.11.14 Γ(z+a)Γ(z+b)(z+a+b-12)a-bk=0Hk(a,b)(z+12(a+b-1))2k.

Here

5.11.15 G0(a,b) =1,
G1(a,b) =12(a-b)(a+b-1),
G2(a,b) =112(a-b2)(3(a+b-1)2-(a-b+1)),
5.11.16 H0(a,b) =1,
H1(a,b) =-112(a-b2)(a-b+1),
H2(a,b) =1240(a-b4)(2(a-b+1)+5(a-b+1)2).

In terms of generalized Bernoulli polynomials Bn()(x)24.16(i)), we have for k=0,1,,

5.11.17 Gk(a,b) =(a-bk)Bk(a-b+1)(a),
5.11.18 Hk(a,b) =(a-b2k)B2k(a-b+1)(a-b+12).

For realistic error bounds in (5.11.14) see Frenzen (1987, 1992). See also Burić and Elezović (2011).

Lastly, and again if z in the sector |phz|π-δ (<π), then

5.11.19 Γ(z+a)Γ(z+b)Γ(z+c)k=0(-1)k(c-a)k(c-b)kk!Γ(a+b-c+z-k).

For the error term in (5.11.19) in the case z=x(>0) and c=1, see Olver (1995).