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

§28.15 Expansions for Small q

  1. §28.15(i) Eigenvalues λν(q)
  2. §28.15(ii) Solutions meν(z,q)

§28.15(i) Eigenvalues λν(q)

28.15.1 λν(q)=ν2+12(ν21)q2+5ν2+732(ν21)3(ν24)q4+9ν4+58ν2+2964(ν21)5(ν24)(ν29)q6+.

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

28.15.2 aν2q2a(ν+2)2q2a(ν+4)2=q2a(ν2)2q2a(ν4)2.

§28.15(ii) Solutions meν(z,q)

28.15.3 meν(z,q)=eiνzq4(1ν+1ei(ν+2)z1ν1ei(ν2)z)+q232(1(ν+1)(ν+2)ei(ν+4)z+1(ν1)(ν2)ei(ν4)z2(ν2+1)(ν21)2eiνz)+;

compare §28.6(ii).