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28 Mathieu Functions and Hill’s EquationModified Mathieu Functions

§28.25 Asymptotic Expansions for Large z

For fixed h(0) and fixed ν,

28.25.1 Mν(3,4)(z,h)e±i(2hcoshz-(12ν+14)π)(πh(coshz+1))12m=0Dm±(4ih(coshz+1))m,

where the coefficients are given by

28.25.2 D-1± =0,
D0± =1,


28.25.3 (m+1)Dm+1±+((m+12)2±(m+14)8ih+2h2-a)Dm±±(m-12)(8ihm)Dm-1±=0,

The upper signs correspond to Mν(3)(z,h) and the lower signs to Mν(4)(z,h). The expansion (28.25.1) is valid for Mν(3)(z,h) when

28.25.4 z+,

and for Mν(4)(z,h) when

28.25.5 z+,

where δ again denotes an arbitrary small positive constant.

For proofs and generalizations see Meixner and Schäfke (1954, §2.63).