# §28.15 Expansions for Small $q$

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

 28.15.1 $\mathop{\lambda_{\nu}\/}\nolimits\!\left(q\right)=\nu^{2}+\frac{1}{2(\nu^{2}-1% )}q^{2}+\frac{5\nu^{2}+7}{32(\nu^{2}-1)^{3}(\nu^{2}-4)}q^{4}+\frac{9\nu^{4}+58% \nu^{2}+29}{64(\nu^{2}-1)^{5}(\nu^{2}-4)(\nu^{2}-9)}q^{6}+\cdots.$ Symbols: $\mathop{\lambda_{\NVar{\nu+2n}}\/}\nolimits\!\left(\NVar{q}\right)$: 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)

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

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

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

 28.15.3 $\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(z,q\right)=e^{\mathrm{i}\nu z}-% \frac{q}{4}\left(\frac{1}{\nu+1}e^{\mathrm{i}(\nu+2)z}-\frac{1}{\nu-1}e^{% \mathrm{i}(\nu-2)z}\right)+\frac{q^{2}}{32}\left(\frac{1}{(\nu+1)(\nu+2)}e^{% \mathrm{i}(\nu+4)z}+\frac{1}{(\nu-1)(\nu-2)}e^{\mathrm{i}(\nu-4)z}-\frac{2(\nu% ^{2}+1)}{(\nu^{2}-1)^{2}}e^{\mathrm{i}\nu z}\right)+\cdots;$ Symbols: $\mathop{\mathrm{me}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $\mathrm{e}$: 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 See also: Annotations for 28.15(ii)

compare §28.6(ii).