As $z\to \mathrm{\infty }$ in the sector $|\mathrm{ph}z|\le \pi -\delta$,

and

For the Bernoulli numbers $B_{2k}$, see §24.2(i).

With the same conditions,

where

Also,

where $a_0=\frac{1}{2}\sqrt{2}$ and

The scaled gamma function ${\mathrm{\Gamma }}^{\ast }\left(z\right)$ is defined in (5.11.3) and its main property is ${\mathrm{\Gamma }}^{\ast }\left(z\right)\sim 1$ as $z\to \mathrm{\infty }$ in the sector $|\mathrm{ph}z|\le \pi -\delta$. Wrench (1968) gives exact values of $g_k$ up to $g_{20}$. Spira (1971) corrects errors in Wrench’s results and also supplies exact and 45D values of $g_k$ for $k=21,22,\mathrm{\dots },30$. For explicit formulas for $g_k$ in terms of Stirling numbers see Nemes (2013a), and for asymptotic expansions of $g_k$ as $k\to \mathrm{\infty }$ see Boyd (1994) and Nemes (2015a).

If the sums in the expansions (5.11.1) and (5.11.2) are terminated at $k=n-1$ ($k\ge 0$) 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 ${\sec }^{2n}\left(\frac{1}{2}\mathrm{ph}z\right)$ for (5.11.1), and ${\sec }^{2n+1}\left(\frac{1}{2}\mathrm{ph}z\right)$ 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

Then

where $\zeta \left(K\right)$ is as in Chapter 25. In the case $K=1$ the factor $1+\zeta \left(K\right)$ is replaced with 4. For this result and a similar bound for the sector $\frac{1}{2}\pi \le \mathrm{ph}z\le \pi$ 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).

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

If $z\to \mathrm{\infty }$ in the sector $|\mathrm{ph}z|\le \pi -\delta$, then

Here

In terms of generalized Bernoulli polynomials $B_n^{(\mathrm{\ell })}\left(x\right)$ (§24.16(i)), we have for $k=0,1,\mathrm{\dots }$,

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

Lastly, and again if $z\to \mathrm{\infty }$ in the sector $|\mathrm{ph}z|\le \pi -\delta$, then

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