As in the sector ,
For the Bernoulli numbers , see §24.2(i).
With the same conditions,
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
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
In this subsection , , and are real or complex constants.
If in the sector (), then
Also, with the added condition ,
In terms of generalized Bernoulli polynomials (§24.16(i)), we have for ,