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28 Mathieu Functions and Hill’s EquationModified Mathieu Functions

§28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

Throughout this section ε0=2 and εs=1, s=1,2,3,.

With 𝒞μ(j), cnν(q), Anm(q), and Bnm(q) as in §28.23,

28.24.1 c2nν(h2)Mν(j)(z,h)==-(-1)c2ν(h2)J-n(he-z)𝒞ν+n+(j)(hez),

where j=1,2,3,4 and n.

In the case when ν is an integer,

28.24.2 εsMc2m(j)(z,h) =(-1)m=0(-1)A22m(h2)A2s2m(h2)(J-s(he-z)𝒞+s(j)(hez)+J+s(he-z)𝒞-s(j)(hez)),
28.24.3 Mc2m+1(j)(z,h) =(-1)m=0(-1)A2+12m+1(h2)A2s+12m+1(h2)(J-s(he-z)𝒞+s+1(j)(hez)+J+s+1(he-z)𝒞-s(j)(hez)),
28.24.4 Ms2m+1(j)(z,h) =(-1)m=0(-1)B2+12m+1(h2)B2s+12m+1(h2)(J-s(he-z)𝒞+s+1(j)(hez)-J+s+1(he-z)𝒞-s(j)(hez)),
28.24.5 Ms2m+2(j)(z,h) =(-1)m=0(-1)B2+22m+2(h2)B2s+22m+2(h2)(J-s(he-z)𝒞+s+2(j)(hez)-J+s+2(he-z)𝒞-s(j)(hez)),

where j=1,2,3,4, and s=0,1,2,.

Also, with In and Kn denoting the modified Bessel functions (§10.25(ii)), and again with s=0,1,2,,

28.24.6 εsIe2m(z,h) =(-1)s=0(-1)A22m(h2)A2s2m(h2)(I-s(he-z)I+s(hez)+I+s(he-z)I-s(hez)),
28.24.7 Io2m+2(z,h) =(-1)s=0(-1)B2+22m+2(h2)B2s+22m+2(h2)(I-s(he-z)I+s+2(hez)-I+s+2(he-z)I-s(hez)),
28.24.8 Ie2m+1(z,h) =(-1)s=0(-1)B2+12m+1(h2)B2s+12m+1(h2)(I-s(he-z)I+s+1(hez)+I+s+1(he-z)I-s(hez)),
28.24.9 Io2m+1(z,h) =(-1)s=0(-1)A2+12m+1(h2)A2s+12m+1(h2)(I-s(he-z)I+s+1(hez)-I+s+1(he-z)I-s(hez)),
28.24.10 εsKe2m(z,h) ==0A22m(h2)A2s2m(h2)(I-s(he-z)K+s(hez)+I+s(he-z)K-s(hez)),
28.24.11 Ko2m+2(z,h) ==0B2+22m+2(h2)B2s+22m+2(h2)(I-s(he-z)K+s+2(hez)-I+s+2(he-z)K-s(hez)),
28.24.12 Ke2m+1(z,h) ==0B2+12m+1(h2)B2s+12m+1(h2)(I-s(he-z)K+s+1(hez)-I+s+1(he-z)K-s(hez)),
28.24.13 Ko2m+1(z,h) ==0A2+12m+1(h2)A2s+12m+1(h2)(I-s(he-z)K+s+1(hez)+I+s+1(he-z)K-s(hez)).

The expansions (28.24.1)–(28.24.13) converge absolutely and uniformly on compact sets of the z-plane.

For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).