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

§28.15 Expansions for Small q

Contents

§28.15(i) Eigenvalues λν(q)

28.15.1 λν(q)=ν2+12(ν2-1)q2+5ν2+732(ν2-1)3(ν2-4)q4+9ν4+58ν2+2964(ν2-1)5(ν2-4)(ν2-9)q6+.

Higher coefficients can be found by equating powers of q in the following continued-fraction equation, with a=λν(q):

28.15.2 a-ν2-q2a-(ν+2)2-q2a-(ν+4)2-=q2a-(ν-2)2-q2a-(ν-4)2-.

§28.15(ii) Solutions meν(z,q)

28.15.3 meν(z,q)=eiνz-q4(1ν+1ei(ν+2)z-1ν-1ei(ν-2)z)+q232(1(ν+1)(ν+2)ei(ν+4)z+1(ν-1)(ν-2)ei(ν-4)z-2(ν2+1)(ν2-1)2eiνz)+;

compare §28.6(ii).