§28.15 Expansions for Small $q$

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

 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

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

 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).