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11 Struve and Related FunctionsRelated Functions

§11.9 Lommel Functions


§11.9(i) Definitions

The inhomogeneous Bessel differential equation

11.9.1 d2wdz2+1zdwdz+(1-ν2z2)w=zμ-1

can be regarded as a generalization of (11.2.7). Provided that μ±ν-1,-3,-5,, (11.9.1) has the general solution

11.9.2 w=sμ,ν(z)+AJν(z)+BYν(z),

where A, B are arbitrary constants, sμ,ν(z) is the Lommel function defined by

11.9.3 sμ,ν(z)=zμ+1k=0(-1)kz2kak+1(μ,ν),


11.9.4 ak(μ,ν)=m=1k((μ+2m-1)2-ν2)=4k(μ-ν+12)k(μ+ν+12)k,

Another solution of (11.9.1) that is defined for all values of μ and ν is Sμ,ν(z), where

11.9.5 Sμ,ν(z)=sμ,ν(z)+2μ-1Γ(12μ+12ν+12)Γ(12μ-12ν+12)×(sin(12(μ-ν)π)Jν(z)-cos(12(μ-ν)π)Yν(z)),

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

Reflection Formulas

11.9.6 sμ,-ν(z) =sμ,ν(z),
Sμ,-ν(z) =Sμ,ν(z).

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

§11.9(ii) Expansions in Series of Bessel Functions

When μ±ν-1,-2,-3,,

11.9.7 sμ,ν(z)=2μ+1k=0(2k+μ+1)Γ(k+μ+1)k!(2k+μ-ν+1)(2k+μ+ν+1)J2k+μ+1(z),
11.9.8 sμ,ν(z)=2(μ+ν-1)/2Γ(12μ+12ν+12)z(μ+1-ν)/2×k=0(12z)kk!(2k+μ-ν+1)Jk+12(μ+ν+1)(z).

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

§11.9(iii) Asymptotic Expansion

For fixed μ and ν,

11.9.9 Sμ,ν(z)zμ-1k=0(-1)kak(-μ,ν)z-2k,
z, |phz|π-δ(<π).

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

For uniform asymptotic expansions, for large ν and fixed μ=-1,0,1,2,, 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).

§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 sμ,ν(x) 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).