Digital Library of Mathematical Functions
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28 Mathieu Functions and Hill’s EquationMathieu Functions of Noninteger Order

§28.15 Expansions for Small q

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§28.15(i) Eigenvalues \mathop{\lambda_{{\nu}}\/}\nolimits\!\left(q\right)

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Notes:
See Meixner and Schäfke (1954, pp. 119 and 117).
Keywords:
Mathieu’s equation
Referenced by:
§28.34(ii)
Permalink:
http://dlmf.nist.gov/28.15.i
28.15.1\mathop{\lambda_{{\nu}}\/}\nolimits\!\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:
\mathop{\lambda_{{\nu+2n}}\/}\nolimits\!\left(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

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

28.15.2a-\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:
§28.34(ii)
Permalink:
http://dlmf.nist.gov/28.15.E2
Encodings:
TeX, pMML, png

§28.15(ii) Solutions \mathop{\mathrm{me}_{{\nu}}\/}\nolimits(z,q)

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Notes:
See Meixner and Schäfke (1954, p. 122).
Keywords:
Mathieu functions
Referenced by:
§28.34(iii)
Permalink:
http://dlmf.nist.gov/28.15.ii
28.15.3\mathop{\mathrm{me}_{{\nu}}\/}\nolimits\!\left(z,q\right)=e^{{i\nu z}}-\frac{q% }{4}\left(\frac{1}{\nu+1}e^{{i(\nu+2)z}}-\frac{1}{\nu-1}e^{{i(\nu-2)z}}\right)% +\frac{q^{2}}{32}\left(\frac{1}{(\nu+1)(\nu+2)}e^{{i(\nu+4)z}}+\frac{1}{(\nu-1% )(\nu-2)}e^{{i(\nu-4)z}}-\frac{2(\nu^{2}+1)}{(\nu^{2}-1)^{2}}e^{{i\nu z}}% \right)+\cdots;
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Symbols:
\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right): Mathieu function, e: base of exponential function, 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

compare §28.6(ii).

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