# §28.6 Expansions for Small $q$

## §28.6(i) Eigenvalues

Leading terms of the power series for $a_{m}\left(q\right)$ and $b_{m}\left(q\right)$ for $m\leq 6$ are:

 28.6.1 $\displaystyle a_{0}\left(q\right)$ $\displaystyle=-\tfrac{1}{2}q^{2}+\tfrac{7}{128}q^{4}-\tfrac{29}{2304}q^{6}+% \tfrac{68687}{188\;74368}q^{8}+\cdots,$ ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation and $q=h^{2}$: parameter A&S Ref: 20.2.25 Referenced by: §28.6(i) Permalink: http://dlmf.nist.gov/28.6.E1 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28 28.6.2 $\displaystyle a_{1}\left(q\right)$ $\displaystyle=1+q-\tfrac{1}{8}q^{2}-\tfrac{1}{64}q^{3}-\tfrac{1}{1536}q^{4}+% \tfrac{11}{36864}q^{5}+\tfrac{49}{5\;89824}q^{6}+\tfrac{55}{94\;37184}q^{7}-% \tfrac{83}{353\;89440}q^{8}+\cdots,$ ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation and $q=h^{2}$: parameter Permalink: http://dlmf.nist.gov/28.6.E2 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28 28.6.3 $\displaystyle b_{1}\left(q\right)$ $\displaystyle=1-q-\tfrac{1}{8}q^{2}+\tfrac{1}{64}q^{3}-\tfrac{1}{1536}q^{4}-% \tfrac{11}{36864}q^{5}+\tfrac{49}{5\;89824}q^{6}-\tfrac{55}{94\;37184}q^{7}-% \tfrac{83}{353\;89440}q^{8}+\cdots,$ ⓘ Symbols: $b_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation and $q=h^{2}$: parameter Permalink: http://dlmf.nist.gov/28.6.E3 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28 28.6.4 $\displaystyle a_{2}\left(q\right)$ $\displaystyle=4+\tfrac{5}{12}q^{2}-\tfrac{763}{13824}q^{4}+\tfrac{10\;02401}{7% 96\;26240}q^{6}-\tfrac{16690\;68401}{45\;86471\;42400}q^{8}+\cdots,$ ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation and $q=h^{2}$: parameter Permalink: http://dlmf.nist.gov/28.6.E4 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28 28.6.5 $\displaystyle b_{2}\left(q\right)$ $\displaystyle=4-\tfrac{1}{12}q^{2}+\tfrac{5}{13824}q^{4}-\tfrac{289}{796\;2624% 0}q^{6}+\tfrac{21391}{45\;86471\;42400}q^{8}+\cdots,$ ⓘ Symbols: $b_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation and $q=h^{2}$: parameter Permalink: http://dlmf.nist.gov/28.6.E5 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28 28.6.6 $\displaystyle a_{3}\left(q\right)$ $\displaystyle=9+\tfrac{1}{16}q^{2}+\tfrac{1}{64}q^{3}+\tfrac{13}{20480}q^{4}-% \tfrac{5}{16384}q^{5}-\tfrac{1961}{235\;92960}q^{6}-\tfrac{609}{1048\;57600}q^% {7}+\cdots,$ ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation and $q=h^{2}$: parameter Permalink: http://dlmf.nist.gov/28.6.E6 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28 28.6.7 $\displaystyle b_{3}\left(q\right)$ $\displaystyle=9+\tfrac{1}{16}q^{2}-\tfrac{1}{64}q^{3}+\tfrac{13}{20480}q^{4}+% \tfrac{5}{16384}q^{5}-\tfrac{1961}{235\;92960}q^{6}+\tfrac{609}{1048\;57600}q^% {7}+\cdots,$ ⓘ Symbols: $b_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation and $q=h^{2}$: parameter Permalink: http://dlmf.nist.gov/28.6.E7 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28 28.6.8 $\displaystyle a_{4}\left(q\right)$ $\displaystyle=16+\tfrac{1}{30}q^{2}+\tfrac{433}{8\;64000}q^{4}-\tfrac{5701}{27% 216\;00000}q^{6}+\cdots,$ ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation and $q=h^{2}$: parameter Permalink: http://dlmf.nist.gov/28.6.E8 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28 28.6.9 $\displaystyle b_{4}\left(q\right)$ $\displaystyle=16+\tfrac{1}{30}q^{2}-\tfrac{317}{8\;64000}q^{4}+\tfrac{10049}{2% 7216\;00000}q^{6}+\cdots,$ ⓘ Symbols: $b_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation and $q=h^{2}$: parameter Permalink: http://dlmf.nist.gov/28.6.E9 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28 28.6.10 $\displaystyle a_{5}\left(q\right)$ $\displaystyle=25+\tfrac{1}{48}q^{2}+\tfrac{11}{7\;74144}q^{4}+\tfrac{1}{1\;474% 56}q^{5}+\tfrac{37}{8918\;13888}q^{6}+\cdots,$ ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation and $q=h^{2}$: parameter Permalink: http://dlmf.nist.gov/28.6.E10 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28 28.6.11 $\displaystyle b_{5}\left(q\right)$ $\displaystyle=25+\tfrac{1}{48}q^{2}+\tfrac{11}{7\;74144}q^{4}-\tfrac{1}{1\;474% 56}q^{5}+\tfrac{37}{8918\;13888}q^{6}+\cdots,$ ⓘ Symbols: $b_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation and $q=h^{2}$: parameter Permalink: http://dlmf.nist.gov/28.6.E11 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28 28.6.12 $\displaystyle a_{6}\left(q\right)$ $\displaystyle=36+\tfrac{1}{70}q^{2}+\tfrac{187}{439\;04000}q^{4}+\tfrac{67\;43% 617}{9293\;59872\;00000}q^{6}+\cdots,$ ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation and $q=h^{2}$: parameter Permalink: http://dlmf.nist.gov/28.6.E12 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28 28.6.13 $\displaystyle b_{6}\left(q\right)$ $\displaystyle=36+\tfrac{1}{70}q^{2}+\tfrac{187}{439\;04000}q^{4}-\tfrac{58\;61% 633}{9293\;59872\;00000}q^{6}+\cdots.$ ⓘ Symbols: $b_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation and $q=h^{2}$: parameter Permalink: http://dlmf.nist.gov/28.6.E13 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28

Leading terms of the of the power series for $m=7,8,9,\dots$ are:

 28.6.14 $\rselection{a_{m}\left(q\right)\\ b_{m}\left(q\right)}=m^{2}+\frac{1}{2(m^{2}-1)}q^{2}+\frac{5m^{2}+7}{32(m^{2}-% 1)^{3}(m^{2}-4)}q^{4}+\frac{9m^{4}+58m^{2}+29}{64(m^{2}-1)^{5}(m^{2}-4)(m^{2}-% 9)}q^{6}+\cdots.$ ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation, $b_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation, $m$: integer and $q=h^{2}$: parameter A&S Ref: 20.2.26 Referenced by: §28.6(i), §28.6(i), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii) Permalink: http://dlmf.nist.gov/28.6.E14 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28

For more details on these expansions and recurrence relations for the coefficients see Frenkel and Portugal (2001, §2).

The coefficients of the power series of $a_{2n}\left(q\right)$, $b_{2n}\left(q\right)$ and also $a_{2n+1}\left(q\right)$, $b_{2n+1}\left(q\right)$ are the same until the terms in $q^{2n-2}$ and $q^{2n}$, respectively. Then

 28.6.15 $a_{m}\left(q\right)-b_{m}\left(q\right)=\frac{2q^{m}}{\left(2^{m-1}(m-1)!% \right)^{2}}\left(1+O\left(q^{2}\right)\right).$

Higher coefficients in the foregoing series can be found by equating coefficients in the following continued-fraction equations:

 28.6.16 $a-(2n)^{2}-\cfrac{q^{2}}{a-(2n-2)^{2}-\cfrac{q^{2}}{a-(2n-4)^{2}-}}\cdots% \cfrac{q^{2}}{a-2^{2}-\cfrac{2q^{2}}{a}}=-\cfrac{q^{2}}{(2n+2)^{2}-a-\cfrac{q^% {2}}{(2n+4)^{2}-a-\cdots}},$ $a=a_{2n}\left(q\right)$, ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation, $q=h^{2}$: parameter, $n$: integer and $a$: parameter Referenced by: item (e) Permalink: http://dlmf.nist.gov/28.6.E16 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28
 28.6.17 $a-(2n+1)^{2}-\cfrac{q^{2}}{a-(2n-1)^{2}-}\cdots\cfrac{q^{2}}{a-3^{2}-\cfrac{q^% {2}}{a-1^{2}-q}}=-\cfrac{q^{2}}{(2n+3)^{2}-a-\cfrac{q^{2}}{(2n+5)^{2}-a-\cdots% }},$ $a=a_{2n+1}\left(q\right)$, ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation, $q=h^{2}$: parameter, $n$: integer and $a$: parameter Permalink: http://dlmf.nist.gov/28.6.E17 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28
 28.6.18 $a-(2n+1)^{2}-\cfrac{q^{2}}{a-(2n-1)^{2}-}\cdots\cfrac{q^{2}}{a-3^{2}-\cfrac{q^% {2}}{a-1^{2}+q}}=-\cfrac{q^{2}}{(2n+3)^{2}-a-\cfrac{q^{2}}{(2n+5)^{2}-a-\cdots% }},$ $a=b_{2n+1}\left(q\right)$, ⓘ Symbols: $b_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation, $q=h^{2}$: parameter, $n$: integer and $a$: parameter Permalink: http://dlmf.nist.gov/28.6.E18 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28
 28.6.19 $a-(2n+2)^{2}-\cfrac{q^{2}}{a-(2n)^{2}-\cfrac{q^{2}}{a-(2n-2)^{2}-}}\cdots% \cfrac{q^{2}}{a-2^{2}}=-\cfrac{q^{2}}{(2n+4)^{2}-a-\cfrac{q^{2}}{(2n+6)^{2}-a-% \cdots}},$ $a=b_{2n+2}\left(q\right)$. ⓘ Symbols: $b_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation, $q=h^{2}$: parameter, $n$: integer and $a$: parameter Referenced by: item (e) Permalink: http://dlmf.nist.gov/28.6.E19 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28

Numerical values of the radii of convergence $\rho_{n}^{(j)}$ of the power series (28.6.1)–(28.6.14) for $n=0,1,\dots,9$ are given in Table 28.6.1. Here $j=1$ for $a_{2n}\left(q\right)$, $j=2$ for $b_{2n+2}\left(q\right)$, and $j=3$ for $a_{2n+1}\left(q\right)$ and $b_{2n+1}\left(q\right)$. (Table 28.6.1 is reproduced from Meixner et al. (1980, §2.4).)

It is conjectured that for large $n$, the radii increase in proportion to the square of the eigenvalue number $n$; see Meixner et al. (1980, §2.4). It is known that

 28.6.20 $\liminf_{n\to\infty}\frac{\rho_{n}^{(j)}}{n^{2}}\geq kk^{\prime}(K\left(k% \right))^{2}=2.04183\;4\dots,$

where $k$ is the unique root of the equation $2E\left(k\right)=K\left(k\right)$ in the interval $(0,1)$, and $k^{\prime}=\sqrt{1-k^{2}}$. For $E\left(k\right)$ and $K\left(k\right)$ see §19.2(ii).

## §28.6(ii) Functions $\operatorname{ce}_{n}$ and $\operatorname{se}_{n}$

Leading terms of the power series for the normalized functions are:

 28.6.21 $\displaystyle 2^{\ifrac{1}{2}}\operatorname{ce}_{0}\left(z,q\right)$ $\displaystyle=1-\tfrac{1}{2}q\cos 2z+\tfrac{1}{32}q^{2}\left(\cos 4z-2\right)-% \tfrac{1}{128}q^{3}\left(\tfrac{1}{9}\cos 6z-11\cos 2z\right)+\cdots,$ ⓘ Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $\cos\NVar{z}$: cosine function, $q=h^{2}$: parameter and $z$: complex variable A&S Ref: 20.2.27 Referenced by: §28.6(ii) Permalink: http://dlmf.nist.gov/28.6.E21 Encodings: TeX, pMML, png See also: Annotations for §28.6(ii), §28.6 and Ch.28 28.6.22 $\displaystyle\operatorname{ce}_{1}\left(z,q\right)$ $\displaystyle=\cos z-\tfrac{1}{8}q\cos 3z+\tfrac{1}{128}q^{2}\left(\tfrac{2}{3% }\cos 5z-2\cos 3z-\cos z\right)-\tfrac{1}{1024}q^{3}\left(\tfrac{1}{9}\cos 7z-% \tfrac{8}{9}\cos 5z-\tfrac{1}{3}\cos 3z+2\cos z\right)+\cdots,$ 28.6.23 $\displaystyle\operatorname{se}_{1}\left(z,q\right)$ $\displaystyle=\sin z-\tfrac{1}{8}q\sin 3z+\tfrac{1}{128}q^{2}\left(\tfrac{2}{3% }\sin 5z+2\sin 3z-\sin z\right)-\tfrac{1}{1024}q^{3}\left(\tfrac{1}{9}\sin 7z+% \tfrac{8}{9}\sin 5z-\tfrac{1}{3}\sin 3z-2\sin z\right)+\cdots,$ 28.6.24 $\displaystyle\operatorname{ce}_{2}\left(z,q\right)$ $\displaystyle=\cos 2z-\tfrac{1}{4}q\left(\tfrac{1}{3}\cos 4z-1\right)+\tfrac{1% }{128}q^{2}\left(\tfrac{1}{3}\cos 6z-\tfrac{76}{9}\cos 2z\right)+\cdots,$ 28.6.25 $\displaystyle\operatorname{se}_{2}\left(z,q\right)$ $\displaystyle=\sin 2z-\tfrac{1}{12}q\sin 4z+\tfrac{1}{128}q^{2}\left(\tfrac{1}% {3}\sin 6z-\tfrac{4}{9}\sin 2z\right)+\cdots.$

For $m=3,4,5,\dots$,

 28.6.26 $\operatorname{ce}_{m}\left(z,q\right)=\cos mz-\frac{q}{4}\left(\frac{1}{m+1}% \cos(m+2)z-\frac{1}{m-1}\cos(m-2)z\right)+\frac{q^{2}}{32}\left(\frac{1}{(m+1)% (m+2)}\cos(m+4)z+\frac{1}{(m-1)(m-2)}\cos(m-4)z-\frac{2(m^{2}+1)}{(m^{2}-1)^{2% }}\cos mz\right)+\cdots.$ ⓘ Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $\cos\NVar{z}$: cosine function, $m$: integer, $q=h^{2}$: parameter and $z$: complex variable A&S Ref: 20.2.28 (in slightly different form) Referenced by: §28.6(ii), §28.6(ii), §28.6(ii), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii) Permalink: http://dlmf.nist.gov/28.6.E26 Encodings: TeX, pMML, png See also: Annotations for §28.6(ii), §28.6 and Ch.28

For the corresponding expansions of $\operatorname{se}_{m}\left(z,q\right)$ for $m=3,4,5,\dots$ change $\cos$ to $\sin$ everywhere in (28.6.26). For more details on these expansions and recurrence relations for the coefficients see Frenkel and Portugal (2001, §2).’

The radii of convergence of the series (28.6.21)–(28.6.26) are the same as the radii of the corresponding series for $a_{n}\left(q\right)$ and $b_{n}\left(q\right)$; compare Table 28.6.1 and (28.6.20).