Source: https://dlmf.nist.gov/5.11
Timestamp: 2019-04-20 13:21:06+00:00

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See Olver (1997b, pp. 87–88 and 293–295) for (5.11.1)–(5.11.3) and (5.11.5)–(5.11.6). (5.11.7) and (5.11.9) are derived from (5.11.3). For (5.11.8) see Rowe (1931) .
A reference to Rowe (1931) was added in the Notes above. References to Nemes (2013a, 2015a) were added after (5.11.6). The reference to Nemes (2013c) that was added in Release 1.0.10 was deleted and moved in revised form to appear after (5.11.8).
The reference to Nemes (2013c) was added at the end of (5.11.2).
To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to Ln⁡Γ⁡(z), where Ln is the general logarithm. Originally ln⁡Γ⁡(z) was used, where ln is the principal branch of the logarithm.
For the Bernoulli numbers B2⁢k, see §24.2(i).
An unnecessary bracket around the sum was removed.
g6 =52 468197 52467 96800.
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 explicit formulas for gk in terms of Stirling numbers see Nemes (2013a) , and for asymptotic expansions of gk as k→∞ see Boyd (1994) and Nemes (2015a) .
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) ).
Originally h in (5.11.8) was unnecessarily restricted to lie in the interval [0,1]. In fact, h may lie anywhere in the complex plane.
To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to Ln⁡Γ⁡(z+h), where Ln is the general logarithm. Originally ln⁡Γ⁡(z+h) was used, where ln is the principal branch of the logarithm.
where h(∈ℂ) is fixed, and Bk⁡(h) is the Bernoulli polynomial defined in §24.2(i). For similar results including a convergent factorial series see, Nemes (2013c) .
uniformly for bounded real values of x.
The reference to Nemes (2013b) that was added in Release 1.0.10 was deleted and moved in revised form to appear at the end of the first paragraph of this subsection.
A reference to Nemes (2013b) has been added at the end of this section.
A sentence dealing with the special case K=1 in (5.11.11) has been added.
If the sums in the expansions (5.11.1) and (5.11.2) are terminated at k=n-1 (k≥0) 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 sec2⁢n⁡(12⁢ph⁡z) for (5.11.1), and sec2⁢n+1⁡(12⁢ph⁡z) for (5.11.2), times the first neglected terms. For error bounds for (5.11.8) and an exponentially-improved extension, see Nemes (2013b) .
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 12⁢π≤ph⁡z≤π 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) .
See Temme (1996b, pp. 67–68) , Olver (1997b, p. 119) , Paris and Kaminski (2001, pp. 50–54) , and Fields (1966) .
In this subsection a, b, and c are real or complex constants.
The previous constraint ℜ⁡(b-a)>0 was removed, see Fields (1966, (3)) .
For realistic error bounds in (5.11.14) see Frenzen (1987, 1992) . See also Burić and Elezović (2011) .
For the error term in (5.11.19) in the case z=x(>0) and c=1, see Olver (1995) .

References: §24
 §12
 §6
 §24
 §12
 §6