§8.11 Asymptotic Approximations and Expansions

Define

Then as $z\to \mathrm{\infty }$ with $a$ and $n$ fixed

where $\delta$ denotes an arbitrary small positive constant.

If $a$ is real and $z$ ($=x$) is positive, then $R_n(a,x)$ is bounded in absolute value by the first neglected term $u_n/x^n$ and has the same sign provided that $n\ge a-1$. For bounds on $R_n(a,z)$ when $a$ is real and $z$ is complex see Olver (1997b, pp. 109–112). For an exponentially-improved asymptotic expansion (§2.11(iii)) see Olver (1991a).

This expansion is absolutely convergent for all finite $z$, and it can also be regarded as a generalized asymptotic expansion (§2.1(v)) of $\gamma (a,z)$ as $a\to \mathrm{\infty }$ in $|\mathrm{ph}a|\le \pi -\delta$.

Also,

If $z=\lambda a$, with $\lambda$ fixed, then as $a\to \mathrm{\infty }$

where

and for $k=1,2,\mathrm{\dots }$,

Sharp error bounds and an exponentially-improved extension for (8.11.7) can be found in Nemes (2016). This reference also contains explicit formulas for $b_k(\lambda )$ in terms of Stirling numbers and for the case $\lambda >1$ an asymptotic expansion for $b_k(\lambda )$ as $k\to \mathrm{\infty }$.

The expansion (8.11.7) also applies when $a$ is replaced by $-a$, and $|\mathrm{ph}a|\le \frac{3\pi }{2}-\omega -\delta$ with $\omega =\mathrm{ph}(-\lambda +\ln \left(-\lambda \right)+\pi \mathrm{i})$, . For error bounds and an exponentially-improved extension for this later expansion, see Nemes (2015c). In the case that $a=n$, a positive integer, the $z$-region of validity of (8.11.7) is discussed in Ameur and Cronvall (2023).

If $x=a+{(2a)}^{\frac{1}{2}}y$ and $a\to +\mathrm{\infty }$, then

in both cases uniformly with respect to bounded real values of $y$. For Dawson’s integral $F\left(y\right)$ see §7.2(ii). See Tricomi (1950b) for these approximations, together with higher terms and extensions to complex variables. For related expansions involving Hermite polynomials see Pagurova (1965).

As $z\to \mathrm{\infty }$,

For sharp error bounds and an exponentially-improved extension, see Nemes (2016). This reference also contains explicit formulas for the coefficients in terms of Stirling numbers.

For the function $e_n(z)$ defined by (8.4.11),

With $x=1$, an asymptotic expansion of $e_n(nx)/{\mathrm{e}}^{nx}$ follows from (8.11.14) and (8.11.16).

If $S_n(x)$ is defined by

then

As $n\to \mathrm{\infty }$

Also,

uniformly for $x\in (-\mathrm{\infty },1-\delta ]$, with

and $b_k(x)$ as in §8.11(iii).

For (8.11.18) and extensions to complex values of $x$ see Buckholtz (1963). For a uniformly valid expansion for $n\to \mathrm{\infty }$ and $x\in [\delta ,1]$, see Wong (1973b).