# §11.11 Asymptotic Expansions of Anger–Weber Functions

## §11.11(ii) Large , Fixed

If is fixed, and in in such a way that is bounded away from the set of all integers, then

If , then (11.10.29) applies for , and

## §11.11(iii) Large , Fixed

For fixed ,

Also, as ,

and

uniformly for bounded real values of . For the Scorer function see §9.12(i).

All of (11.11.10)–(11.11.17) can be regarded as special cases of two asymptotic expansions given in Olver (1997b, pp. 352–357) for as , one being uniform for , where again denotes an arbitrary small positive constant, and the other being uniform for . (Note that Olver’s definition of omits the factor in (11.10.4).) See also Watson (1944, §10.15).

Lastly, corresponding asymptotic approximations and expansions for and follow from (11.10.15) and (11.10.16) and the corresponding asymptotic expansions for the Bessel functions and ; see §10.19(ii). In particular,