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

§28.15 Expansions for Small q

Contents
  1. §28.15(i) Eigenvalues \lambda_{\nu}\left(q\right)
  2. §28.15(ii) Solutions \operatorname{me}_{\nu}(z,q)

§28.15(i) Eigenvalues \lambda_{\nu}\left(q\right)

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Keywords:
Mathieu’s equation, continued-fraction equations, eigenvalues (or characteristic values), power-series expansions in q
Notes:
See Meixner and Schäfke (1954, pp. 119 and 117).
Referenced by:
item (a)
Permalink:
http://dlmf.nist.gov/28.15.i
See also:
Annotations for §28.15 and Ch.28
28.15.1 \lambda_{\nu}\left(q\right)=\nu^{2}+\frac{1}{2(\nu^{2}-1)}q^{2}+\frac{5\nu^{2}% +7}{32(\nu^{2}-1)^{3}(\nu^{2}-4)}q^{4}+\frac{9\nu^{4}+58\nu^{2}+29}{64(\nu^{2}% -1)^{5}(\nu^{2}-4)(\nu^{2}-9)}q^{6}+\cdots.
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Symbols:
\lambda_{\NVar{\nu+2n}}\left(\NVar{q}\right): eigenvalues of Mathieu equation, q=h^{2}: parameter and \nu: complex parameter
A&S Ref:
20.3.15 (in slightly different form)
Permalink:
http://dlmf.nist.gov/28.15.E1
Encodings:
TeX, pMML, png
See also:
Annotations for §28.15(i), §28.15 and Ch.28

Higher coefficients can be found by equating powers of q in the following continued-fraction equation, with a=\lambda_{\nu}\left(q\right):

28.15.2 a-\nu^{2}-\cfrac{q^{2}}{a-(\nu+2)^{2}-\cfrac{q^{2}}{a-(\nu+4)^{2}-\cdots}}=% \cfrac{q^{2}}{a-(\nu-2)^{2}-\cfrac{q^{2}}{a-(\nu-4)^{2}-\cdots}}.
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Symbols:
q=h^{2}: parameter, \nu: complex parameter and a: parameter
Referenced by:
item (e)
Permalink:
http://dlmf.nist.gov/28.15.E2
Encodings:
TeX, pMML, png
See also:
Annotations for §28.15(i), §28.15 and Ch.28

§28.15(ii) Solutions \operatorname{me}_{\nu}(z,q)

ⓘ
Keywords:
Mathieu functions, power series in q
Notes:
See Meixner and Schäfke (1954, p. 122).
Referenced by:
item (a)
Permalink:
http://dlmf.nist.gov/28.15.ii
See also:
Annotations for §28.15 and Ch.28
28.15.3 \operatorname{me}_{\nu}\left(z,q\right)=e^{\mathrm{i}\nu z}-\frac{q}{4}\left(% \frac{1}{\nu+1}e^{\mathrm{i}(\nu+2)z}-\frac{1}{\nu-1}e^{\mathrm{i}(\nu-2)z}% \right)+\frac{q^{2}}{32}\left(\frac{1}{(\nu+1)(\nu+2)}e^{\mathrm{i}(\nu+4)z}+% \frac{1}{(\nu-1)(\nu-2)}e^{\mathrm{i}(\nu-4)z}-\frac{2(\nu^{2}+1)}{(\nu^{2}-1)% ^{2}}e^{\mathrm{i}\nu z}\right)+\cdots;
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Symbols:
\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right): Mathieu function, \mathrm{e}: base of natural logarithm, \mathrm{i}: imaginary unit, q=h^{2}: parameter, z: complex variable and \nu: complex parameter
A&S Ref:
20.3.17 (only two terms and without normalization)
Permalink:
http://dlmf.nist.gov/28.15.E3
Encodings:
TeX, pMML, png
See also:
Annotations for §28.15(ii), §28.15 and Ch.28

compare §28.6(ii).

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