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§28.15 Expansions for Small $q$

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Keywords:: Fourier coefficients, Mathieu functions, asymptotic forms for small $q$
Permalink:: http://dlmf.nist.gov/28.15
See also:: Annotations for Ch.28

§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		$\lambda_{\nu}\left(q\right)=\nu^{2}+\frac{1}{2(\nu^{2}-1)}q^{2}+\frac{5\nu^{2}% +7}{32(\nu^{2}-1)^{3}(\nu^{2}-4)}q^{4}+\frac{9\nu^{4}+58\nu^{2}+29}{64(\nu^{2}% -1)^{5}(\nu^{2}-4)(\nu^{2}-9)}q^{6}+\cdots.$
ⓘ 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		$a-\nu^{2}-\cfrac{q^{2}}{a-(\nu+2)^{2}-\cfrac{q^{2}}{a-(\nu+4)^{2}-\cdots}}=% \cfrac{q^{2}}{a-(\nu-2)^{2}-\cfrac{q^{2}}{a-(\nu-4)^{2}-\cdots}}.$
ⓘ 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

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28.15.3		$\operatorname{me}_{\nu}\left(z,q\right)=e^{\mathrm{i}\nu z}-\frac{q}{4}\left(% \frac{1}{\nu+1}e^{\mathrm{i}(\nu+2)z}-\frac{1}{\nu-1}e^{\mathrm{i}(\nu-2)z}% \right)+\frac{q^{2}}{32}\left(\frac{1}{(\nu+1)(\nu+2)}e^{\mathrm{i}(\nu+4)z}+% \frac{1}{(\nu-1)(\nu-2)}e^{\mathrm{i}(\nu-4)z}-\frac{2(\nu^{2}+1)}{(\nu^{2}-1)% ^{2}}e^{\mathrm{i}\nu z}\right)+\cdots;$
ⓘ 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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