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28 Mathieu Functions and Hill’s EquationMathieu Functions of Noninteger Order

§28.14 Fourier Series

The Fourier series

28.14.1 meν(z,q) =m=c2mν(q)ei(ν+2m)z,
28.14.2 ceν(z,q) =m=c2mν(q)cos(ν+2m)z,
28.14.3 seν(z,q) =m=c2mν(q)sin(ν+2m)z,

converge absolutely and uniformly on all compact sets in the z-plane. The coefficients satisfy

28.14.4 qc2m+2(a(ν+2m)2)c2m+qc2m2=0,

and the normalization relation

28.14.5 m=(c2mν(q))2=1;

compare (28.12.5). Ambiguities in sign are resolved by (28.14.9) when q=0, and by continuity for other values of q.

The rate of convergence is indicated by

28.14.6 c2mν(q)c2m2ν(q)=q4m2(1+O(1m)),

For changes of sign of ν, q, and m,

28.14.7 c2mν(q) =c2mν(q),
28.14.8 c2mν(q) =(1)mc2mν(q).

When q=0,

28.14.9 c0ν(0) =1,
c2mν(0) =0,

When q0 with m (1) and ν fixed,

28.14.10 c2mν(q)=((1)mqmΓ(ν+1)m! 22mΓ(ν+m+1)+O(qm+2))c0ν(q).