§28.6 Expansions for Small $q$

§28.6(i) Eigenvalues

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

 28.6.1 $\displaystyle\mathop{a_{0}\/}\nolimits\!\left(q\right)$ $\displaystyle=-\tfrac{1}{2}q^{2}+\tfrac{7}{128}q^{4}-\tfrac{29}{2304}q^{6}+% \tfrac{68687}{188\;74368}q^{8}+\cdots,$ Symbols: $\mathop{a_{\NVar{n}}\/}\nolimits\!\left(\NVar{q}\right)$: 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) 28.6.2 $\displaystyle\mathop{a_{1}\/}\nolimits\!\left(q\right)$ $\displaystyle=1+q-\tfrac{1}{8}q^{2}-\tfrac{1}{64}q^{3}-\tfrac{1}{1536}q^{4}+% \tfrac{11}{36864}q^{5}+\tfrac{49}{5\;89824}q^{6}+\tfrac{55}{94\;37184}q^{7}-% \tfrac{83}{353\;89440}q^{8}+\cdots,$ Symbols: $\mathop{a_{\NVar{n}}\/}\nolimits\!\left(\NVar{q}\right)$: 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) 28.6.3 $\displaystyle\mathop{b_{1}\/}\nolimits\!\left(q\right)$ $\displaystyle=1-q-\tfrac{1}{8}q^{2}+\tfrac{1}{64}q^{3}-\tfrac{1}{1536}q^{4}-% \tfrac{11}{36864}q^{5}+\tfrac{49}{5\;89824}q^{6}-\tfrac{55}{94\;37184}q^{7}-% \tfrac{83}{353\;89440}q^{8}+\cdots,$ Symbols: $\mathop{b_{\NVar{n}}\/}\nolimits\!\left(\NVar{q}\right)$: 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) 28.6.4 $\displaystyle\mathop{a_{2}\/}\nolimits\!\left(q\right)$ $\displaystyle=4+\tfrac{5}{12}q^{2}-\tfrac{763}{13824}q^{4}+\tfrac{10\;02401}{7% 96\;26240}q^{6}-\tfrac{16690\;68401}{45\;86471\;42400}q^{8}+\cdots,$ Symbols: $\mathop{a_{\NVar{n}}\/}\nolimits\!\left(\NVar{q}\right)$: 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) 28.6.5 $\displaystyle\mathop{b_{2}\/}\nolimits\!\left(q\right)$ $\displaystyle=4-\tfrac{1}{12}q^{2}+\tfrac{5}{13824}q^{4}-\tfrac{289}{796\;2624% 0}q^{6}+\tfrac{21391}{45\;86471\;42400}q^{8}+\cdots,$ Symbols: $\mathop{b_{\NVar{n}}\/}\nolimits\!\left(\NVar{q}\right)$: 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) 28.6.6 $\displaystyle\mathop{a_{3}\/}\nolimits\!\left(q\right)$ $\displaystyle=9+\tfrac{1}{16}q^{2}+\tfrac{1}{64}q^{3}+\tfrac{13}{20480}q^{4}-% \tfrac{5}{16384}q^{5}-\tfrac{1961}{235\;92960}q^{6}-\tfrac{609}{1048\;57600}q^% {7}+\cdots,$ Symbols: $\mathop{a_{\NVar{n}}\/}\nolimits\!\left(\NVar{q}\right)$: 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) 28.6.7 $\displaystyle\mathop{b_{3}\/}\nolimits\!\left(q\right)$ $\displaystyle=9+\tfrac{1}{16}q^{2}-\tfrac{1}{64}q^{3}+\tfrac{13}{20480}q^{4}+% \tfrac{5}{16384}q^{5}-\tfrac{1961}{235\;92960}q^{6}+\tfrac{609}{1048\;57600}q^% {7}+\cdots,$ Symbols: $\mathop{b_{\NVar{n}}\/}\nolimits\!\left(\NVar{q}\right)$: 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) 28.6.8 $\displaystyle\mathop{a_{4}\/}\nolimits\!\left(q\right)$ $\displaystyle=16+\tfrac{1}{30}q^{2}+\tfrac{433}{8\;64000}q^{4}-\tfrac{5701}{27% 216\;00000}q^{6}+\cdots,$ Symbols: $\mathop{a_{\NVar{n}}\/}\nolimits\!\left(\NVar{q}\right)$: 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) 28.6.9 $\displaystyle\mathop{b_{4}\/}\nolimits\!\left(q\right)$ $\displaystyle=16+\tfrac{1}{30}q^{2}-\tfrac{317}{8\;64000}q^{4}+\tfrac{10049}{2% 7216\;00000}q^{6}+\cdots,$ Symbols: $\mathop{b_{\NVar{n}}\/}\nolimits\!\left(\NVar{q}\right)$: 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) 28.6.10 $\displaystyle\mathop{a_{5}\/}\nolimits\!\left(q\right)$ $\displaystyle=25+\tfrac{1}{48}q^{2}+\tfrac{11}{7\;74144}q^{4}+\tfrac{1}{1\;474% 56}q^{5}+\tfrac{37}{8918\;13888}q^{6}+\cdots,$ Symbols: $\mathop{a_{\NVar{n}}\/}\nolimits\!\left(\NVar{q}\right)$: 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) 28.6.11 $\displaystyle\mathop{b_{5}\/}\nolimits\!\left(q\right)$ $\displaystyle=25+\tfrac{1}{48}q^{2}+\tfrac{11}{7\;74144}q^{4}-\tfrac{1}{1\;474% 56}q^{5}+\tfrac{37}{8918\;13888}q^{6}+\cdots,$ Symbols: $\mathop{b_{\NVar{n}}\/}\nolimits\!\left(\NVar{q}\right)$: 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) 28.6.12 $\displaystyle\mathop{a_{6}\/}\nolimits\!\left(q\right)$ $\displaystyle=36+\tfrac{1}{70}q^{2}+\tfrac{187}{439\;04000}q^{4}+\tfrac{67\;43% 617}{9293\;59872\;00000}q^{6}+\cdots,$ Symbols: $\mathop{a_{\NVar{n}}\/}\nolimits\!\left(\NVar{q}\right)$: 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) 28.6.13 $\displaystyle\mathop{b_{6}\/}\nolimits\!\left(q\right)$ $\displaystyle=36+\tfrac{1}{70}q^{2}+\tfrac{187}{439\;04000}q^{4}-\tfrac{58\;61% 633}{9293\;59872\;00000}q^{6}+\cdots.$ Symbols: $\mathop{b_{\NVar{n}}\/}\nolimits\!\left(\NVar{q}\right)$: 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)

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

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

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

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

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

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

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 $\mathop{a_{2n}\/}\nolimits\!\left(q\right)$, $j=2$ for $\mathop{b_{2n+2}\/}\nolimits\!\left(q\right)$, and $j=3$ for $\mathop{a_{2n+1}\/}\nolimits\!\left(q\right)$ and $\mathop{b_{2n+1}\/}\nolimits\!\left(q\right)$. (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 $\liminf_{n\to\infty}\frac{\rho_{n}^{(j)}}{n^{2}}\geq kk^{\prime}(\mathop{K\/}% \nolimits\!\left(k\right))^{2}=2.04183\;4\dots,$

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

§28.6(ii) Functions $\mathop{\mathrm{ce}_{n}\/}\nolimits$ and $\mathop{\mathrm{se}_{n}\/}\nolimits$

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

 28.6.21 $\displaystyle 2^{\ifrac{1}{2}}\mathop{\mathrm{ce}_{0}\/}\nolimits\!\left(z,q\right)$ $\displaystyle=1-\tfrac{1}{2}q\mathop{\cos\/}\nolimits 2z+\tfrac{1}{32}q^{2}% \left(\mathop{\cos\/}\nolimits 4z-2\right)-\tfrac{1}{128}q^{3}\left(\tfrac{1}{% 9}\mathop{\cos\/}\nolimits 6z-11\mathop{\cos\/}\nolimits 2z\right)+\cdots,$ Symbols: $\mathop{\mathrm{ce}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $\mathop{\cos\/}\nolimits\NVar{z}$: 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) 28.6.22 $\displaystyle\mathop{\mathrm{ce}_{1}\/}\nolimits\!\left(z,q\right)$ $\displaystyle=\mathop{\cos\/}\nolimits z-\tfrac{1}{8}q\mathop{\cos\/}\nolimits 3% z+\tfrac{1}{128}q^{2}\left(\tfrac{2}{3}\mathop{\cos\/}\nolimits 5z-2\mathop{% \cos\/}\nolimits 3z-\mathop{\cos\/}\nolimits z\right)-\tfrac{1}{1024}q^{3}% \left(\tfrac{1}{9}\mathop{\cos\/}\nolimits 7z-\tfrac{8}{9}\mathop{\cos\/}% \nolimits 5z-\tfrac{1}{3}\mathop{\cos\/}\nolimits 3z+2\mathop{\cos\/}\nolimits z% \right)+\cdots,$ 28.6.23 $\displaystyle\mathop{\mathrm{se}_{1}\/}\nolimits\!\left(z,q\right)$ $\displaystyle=\mathop{\sin\/}\nolimits z-\tfrac{1}{8}q\mathop{\sin\/}\nolimits 3% z+\tfrac{1}{128}q^{2}\left(\tfrac{2}{3}\mathop{\sin\/}\nolimits 5z+2\mathop{% \sin\/}\nolimits 3z-\mathop{\sin\/}\nolimits z\right)-\tfrac{1}{1024}q^{3}% \left(\tfrac{1}{9}\mathop{\sin\/}\nolimits 7z+\tfrac{8}{9}\mathop{\sin\/}% \nolimits 5z-\tfrac{1}{3}\mathop{\sin\/}\nolimits 3z-2\mathop{\sin\/}\nolimits z% \right)+\cdots,$ 28.6.24 $\displaystyle\mathop{\mathrm{ce}_{2}\/}\nolimits\!\left(z,q\right)$ $\displaystyle=\mathop{\cos\/}\nolimits 2z-\tfrac{1}{4}q\left(\tfrac{1}{3}% \mathop{\cos\/}\nolimits 4z-1\right)+\tfrac{1}{128}q^{2}\left(\tfrac{1}{3}% \mathop{\cos\/}\nolimits 6z-\tfrac{76}{9}\mathop{\cos\/}\nolimits 2z\right)+\cdots,$ 28.6.25 $\displaystyle\mathop{\mathrm{se}_{2}\/}\nolimits\!\left(z,q\right)$ $\displaystyle=\mathop{\sin\/}\nolimits 2z-\tfrac{1}{12}q\mathop{\sin\/}% \nolimits 4z+\tfrac{1}{128}q^{2}\left(\tfrac{1}{3}\mathop{\sin\/}\nolimits 6z-% \tfrac{4}{9}\mathop{\sin\/}\nolimits 2z\right)+\cdots.$

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

 28.6.26 $\mathop{\mathrm{ce}_{m}\/}\nolimits\!\left(z,q\right)=\mathop{\cos\/}\nolimits mz% -\frac{q}{4}\left(\frac{1}{m+1}\mathop{\cos\/}\nolimits(m+2)z-\frac{1}{m-1}% \mathop{\cos\/}\nolimits(m-2)z\right)+\frac{q^{2}}{32}\left(\frac{1}{(m+1)(m+2% )}\mathop{\cos\/}\nolimits(m+4)z+\frac{1}{(m-1)(m-2)}\mathop{\cos\/}\nolimits(% m-4)z-\frac{2(m^{2}+1)}{(m^{2}-1)^{2}}\mathop{\cos\/}\nolimits mz\right)+\cdots.$ Symbols: $\mathop{\mathrm{ce}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $\mathop{\cos\/}\nolimits\NVar{z}$: 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) Permalink: http://dlmf.nist.gov/28.6.E26 Encodings: TeX, pMML, png See also: Annotations for 28.6(ii)

For the corresponding expansions of $\mathop{\mathrm{se}_{m}\/}\nolimits\!\left(z,q\right)$ for $m=3,4,5,\dots$ change $\mathop{\cos\/}\nolimits$ to $\mathop{\sin\/}\nolimits$ everywhere in (28.6.26).

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