# §11.9 Lommel Functions

## §11.9(i) Definitions

The inhomogeneous Bessel differential equation

can be regarded as a generalization of (11.2.7). Provided that , (11.9.1) has the general solution

where , are arbitrary constants, is the Lommel function defined by

11.9.3

and

Another solution of (11.9.1) that is defined for all values of and is , where

the right-hand side being replaced by its limiting form when is an odd negative integer.

### ¶ Reflection Formulas

For the foregoing results and further information see Watson (1944, §§10.7–10.73) and Babister (1967, §3.16).

## §11.9(iii) Asymptotic Expansion

For fixed and ,

For see (11.9.4). If either of equals an odd positive integer, then the right-hand side of (11.9.9) terminates and represents exactly.

For uniform asymptotic expansions, for large and fixed , of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390).

## §11.9(iv) References

For further information on Lommel functions see Watson (1944, §§10.7–10.75) and Babister (1967, Chapter 3). For descriptive properties of 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 (2000, §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).