As in the sector ,
For the Bernoulli numbers , see §24.2(i).
With the same conditions,
The scaled gamma function is defined in (5.11.3) and its main property is as in the sector . Wrench (1968) gives exact values of up to . Spira (1971) corrects errors in Wrench’s results and also supplies exact and 45D values of for . For explicit formulas for in terms of Stirling numbers see Nemes (2013a), and for asymptotic expansions of as 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)).
Next, and again with the same conditions,
where and are both fixed, and
where is fixed, and is the Bernoulli polynomial defined in §24.2(i). For similar results including a convergent factorial series see, Nemes (2013c).
Lastly, as ,
uniformly for bounded real values of .
If the sums in the expansions (5.11.1) and (5.11.2) are terminated at () and is real and positive, then the remainder terms are bounded in magnitude by the first neglected terms and have the same sign. If is complex, then the remainder terms are bounded in magnitude by for (5.11.1), and for (5.11.2), times the first neglected terms. For error bounds for (5.11.8) and an exponentially-improved extension, see Nemes (2013b).
For the remainder term in (5.11.3) write
where is as in Chapter 25. In the case the factor is replaced with 4. For this result and a similar bound for the sector 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).
In this subsection , , and are real or complex constants.
If in the sector , then
In terms of generalized Bernoulli polynomials (§24.16(i)), we have for ,
For realistic error bounds in (5.11.14) see Frenzen (1987a, 1992). See also Burić and Elezović (2011).