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28 Mathieu Functions and Hill’s EquationModified Mathieu Functions

§28.23 Expansions in Series of Bessel Functions

We use the following notations:

28.23.1 𝒞μ(1) =Jμ,
𝒞μ(2) =Yμ,
𝒞μ(3) =Hμ(1),
𝒞μ(4) =Hμ(2);

compare §10.2(ii). For the coefficients cnν(q) see §28.14. For Anm(q) and Bnm(q) see §28.4.

28.23.2 meν(0,h2)Mν(j)(z,h) =n=-(-1)nc2nν(h2)𝒞ν+2n(j)(2hcoshz),
28.23.3 meν(0,h2)Mν(j)(z,h) =itanhzn=-(-1)n(ν+2n)c2nν(h2)𝒞ν+2n(j)(2hcoshz),

valid for all z when j=1, and for z>0 and |coshz|>1 when j=2,3,4.

28.23.4 meν(12π,h2)Mν(j)(z,h) =eiνπ/2n=-c2nν(h2)𝒞ν+2n(j)(2hsinhz),
28.23.5 meν(12π,h2)Mν(j)(z,h) =ieiνπ/2cothzn=-(ν+2n)c2nν(h2)𝒞ν+2n(j)(2hsinhz),

valid for all z when j=1, and for z>0 and |sinhz|>1 when j=2,3,4.

In the case when ν is an integer

28.23.6 Mc2m(j)(z,h) =(-1)m(ce2m(0,h2))-1=0(-1)A22m(h2)𝒞2(j)(2hcoshz),
28.23.7 Mc2m(j)(z,h) =(-1)m(ce2m(12π,h2))-1=0A22m(h2)𝒞2(j)(2hsinhz),
28.23.8 Mc2m+1(j)(z,h) =(-1)m(ce2m+1(0,h2))-1=0(-1)A2+12m+1(h2)𝒞2+1(j)(2hcoshz),
28.23.9 Mc2m+1(j)(z,h) =(-1)m+1(ce2m+1(12π,h2))-1cothz=0(2+1)A2+12m+1(h2)𝒞2+1(j)(2hsinhz),
28.23.10 Ms2m+1(j)(z,h) =(-1)m(se2m+1(0,h2))-1tanhz=0(-1)(2+1)B2+12m+1(h2)𝒞2+1(j)(2hcoshz),
28.23.11 Ms2m+1(j)(z,h) =(-1)m(se2m+1(12π,h2))-1=0B2+12m+1(h2)𝒞2+1(j)(2hsinhz),
28.23.12 Ms2m+2(j)(z,h) =(-1)m(se2m+2(0,h2))-1tanhz=0(-1)(2+2)B2+22m+2(h2)𝒞2+2(j)(2hcoshz),
28.23.13 Ms2m+2(j)(z,h) =(-1)m+1(se2m+2(12π,h2))-1cothz=0(2+2)B2+22m+2(h2)𝒞2+2(j)(2hsinhz).

When j=1, each of the series (28.23.6)–(28.23.13) converges for all z. When j=2,3,4 the series in the even-numbered equations converge for z>0 and |coshz|>1, and the series in the odd-numbered equations converge for z>0 and |sinhz|>1.

For proofs and generalizations, see Meixner and Schäfke (1954, §§2.62 and 2.64).