§11.11 Asymptotic Expansions of Anger–Weber Functions

Let $F_0(\nu )=G_0(\nu )=1$, and for $k=1,2,3,\mathrm{\dots }$,

Then as $z\to \mathrm{\infty }$ in $|\mathrm{ph}z|\le \pi -\delta$

For sharp error bounds and exponentially-improved extensions, see Nemes (2018).

If $z$ is fixed, and $\nu \to \mathrm{\infty }$ in $|\mathrm{ph}\nu |\le \pi$ in such a way that $\nu$ is bounded away from the set of all integers, then

If $\nu =n(\in \mathbb{Z})$, then (11.10.29) applies for ${𝐉}_n\left(z\right)$, and

as $n\to \pm \mathrm{\infty }$.

For fixed $\lambda (>1)$,

In particular, as $\nu \to \mathrm{\infty }$,

Also, as $\nu \to \mathrm{\infty }$ in $|\mathrm{ph}\nu |\le 2\pi -\delta$,

and

uniformly for bounded complex values of $a$. For the Scorer function $\mathrm{Hi}$ see §9.12(i).

Error bounds for (11.11.8) and (11.11.10) are given in Meijer (1932) and Nemes (2014b, c). The later references also contain exponentially-improved extensions of (11.11.8) and (11.11.10). For an extension of (11.11.17) (and (11.11.16)) into a complete asymptotic expansion, see Nemes (2020).

When $\nu$ is real and positive, all of (11.11.10)–(11.11.17) can be regarded as special cases of two asymptotic expansions given in Olver (1997b, pp. 352–360) for ${𝐀}_{-\nu }\left(\lambda \nu \right)$ as $\nu \to +\mathrm{\infty }$, one being uniform for , and the other being uniform for $\lambda \ge 1$. (Note that Olver’s definition of ${𝐀}_{\nu }\left(z\right)$ omits the factor $1/\pi$ in (11.10.4).) See also Watson (1944, §10.15).

Lastly, corresponding asymptotic approximations and expansions for ${𝐉}_{\nu }\left(\lambda \nu \right)$ and ${𝐄}_{\nu }\left(\lambda \nu \right)$, with or $\lambda >1$, follow from (11.10.15) and (11.10.16) and the corresponding asymptotic expansions for the Bessel functions $J_{\nu }\left(z\right)$ and $Y_{\nu }\left(z\right)$; see §10.19(ii). Furthermore,