§11.9 Lommel Functions

The inhomogeneous Bessel differential equation

can be regarded as a generalization of (11.2.7). Provided that $\mu \pm \nu \ne -1,-3,-5,\mathrm{\dots }$, (11.9.1) has the general solution

where $A$, $B$ are arbitrary constants, $s_{\mu ,\nu }\left(z\right)$ is the Lommel function defined by

and

Another solution of (11.9.1) that is defined for all values of $\mu$ and $\nu$ is $S_{\mu ,\nu }\left(z\right)$, where

the right-hand side being replaced by its limiting form when $\mu \pm \nu$ is an odd negative integer.

When $\mu \pm \nu \ne -1,-2,-3,\mathrm{\dots }$,

For these and further results see Luke (1969b, §9.4.5).

For fixed $\mu$ and $\nu$,

For $a_k(\mu ,\nu )$ see (11.9.4). If either of $\mu \pm \nu$ equals an odd positive integer, then the right-hand side of (11.9.9) terminates and represents $S_{\mu ,\nu }\left(z\right)$ exactly.

For uniform asymptotic expansions, for large $\nu$ and fixed $\mu =-1,0,1,2,\mathrm{\dots }$, of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390). For an error bound for (11.9.9) and an exponentially-improved extension see Nemes (2015b).

For further information on Lommel functions see Watson (1944, §§10.7–10.75) and Babister (1967, Chapter 3). For descriptive properties of $s_{\mu ,\nu }\left(x\right)$ see Steinig (1972).

For collections of integral representations and integrals see Apelblat (1983, §12.17), Babister (1967, p. 85), Erdélyi et al. (1954a, §§4.19 and 5.17), Gradshteyn and Ryzhik (2015, §6.86), Marichev (1983, p. 193), Oberhettinger (1972, pp. 127–128, 168–169, and 188–189), Oberhettinger (1974, §§1.12 and 2.7), Oberhettinger (1990, pp. 105–106 and 191–192), Oberhettinger and Badii (1973, §2.14), Prudnikov et al. (1990, §§1.6 and 2.9), Prudnikov et al. (1992a, §3.34), and Prudnikov et al. (1992b, §3.32).