§28.6 Expansions for Small $q$

Leading terms of the power series for $a_m\left(q\right)$ and $b_m\left(q\right)$ for $m\le 6$ are:

28.6.1		$a_0\left(q\right)$	$=-\frac{1}{2}{q}^2+\frac{7}{128}{q}^4-\frac{29}{2304}{q}^6+\frac{68687}{\mathrm{188\hspace{0.33em}74368}}{q}^8+\mathrm{\cdots },$
ⓘ Symbols: $a_n\left(q\right)$: 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		$a_1\left(q\right)$	$=\begin{array}{l}1+q-\frac{1}{8}{q}^2-\frac{1}{64}{q}^3-\frac{1}{1536}{q}^4+\frac{11}{36864}{q}^5+\frac{49}{\mathrm{5\hspace{0.33em}89824}}{q}^6+\frac{55}{\mathrm{94\hspace{0.33em}37184}}{q}^7-\frac{83}{\mathrm{353\hspace{0.33em}89440}}{q}^8\\ +\mathrm{\cdots },\end{array}$
ⓘ Symbols: $a_n\left(q\right)$: 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		$b_1\left(q\right)$	$=\begin{array}{l}1-q-\frac{1}{8}{q}^2+\frac{1}{64}{q}^3-\frac{1}{1536}{q}^4-\frac{11}{36864}{q}^5+\frac{49}{\mathrm{5\hspace{0.33em}89824}}{q}^6-\frac{55}{\mathrm{94\hspace{0.33em}37184}}{q}^7-\frac{83}{\mathrm{353\hspace{0.33em}89440}}{q}^8\\ +\mathrm{\cdots },\end{array}$
ⓘ Symbols: $b_n\left(q\right)$: 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		$a_2\left(q\right)$	$=4+\frac{5}{12}{q}^2-\frac{763}{13824}{q}^4+\frac{\mathrm{10\hspace{0.33em}02401}}{\mathrm{796\hspace{0.33em}26240}}{q}^6-\frac{\mathrm{16690\hspace{0.33em}68401}}{\mathrm{45\hspace{0.33em}86471\hspace{0.33em}42400}}{q}^8+\mathrm{\cdots },$
ⓘ Symbols: $a_n\left(q\right)$: 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		$b_2\left(q\right)$	$=4-\frac{1}{12}{q}^2+\frac{5}{13824}{q}^4-\frac{289}{\mathrm{796\hspace{0.33em}26240}}{q}^6+\frac{21391}{\mathrm{45\hspace{0.33em}86471\hspace{0.33em}42400}}{q}^8+\mathrm{\cdots },$
ⓘ Symbols: $b_n\left(q\right)$: 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		$a_3\left(q\right)$	$=9+\frac{1}{16}{q}^2+\frac{1}{64}{q}^3+\frac{13}{20480}{q}^4-\frac{5}{16384}{q}^5-\frac{1961}{\mathrm{235\hspace{0.33em}92960}}{q}^6-\frac{609}{\mathrm{1048\hspace{0.33em}57600}}{q}^7+\mathrm{\cdots },$
ⓘ Symbols: $a_n\left(q\right)$: 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		$b_3\left(q\right)$	$=9+\frac{1}{16}{q}^2-\frac{1}{64}{q}^3+\frac{13}{20480}{q}^4+\frac{5}{16384}{q}^5-\frac{1961}{\mathrm{235\hspace{0.33em}92960}}{q}^6+\frac{609}{\mathrm{1048\hspace{0.33em}57600}}{q}^7+\mathrm{\cdots },$
ⓘ Symbols: $b_n\left(q\right)$: 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		$a_4\left(q\right)$	$=16+\frac{1}{30}{q}^2+\frac{433}{\mathrm{8\hspace{0.33em}64000}}{q}^4-\frac{5701}{\mathrm{27216\hspace{0.33em}00000}}{q}^6+\mathrm{\cdots },$
ⓘ Symbols: $a_n\left(q\right)$: 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		$b_4\left(q\right)$	$=16+\frac{1}{30}{q}^2-\frac{317}{\mathrm{8\hspace{0.33em}64000}}{q}^4+\frac{10049}{\mathrm{27216\hspace{0.33em}00000}}{q}^6+\mathrm{\cdots },$
ⓘ Symbols: $b_n\left(q\right)$: 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		$a_5\left(q\right)$	$=25+\frac{1}{48}{q}^2+\frac{11}{\mathrm{7\hspace{0.33em}74144}}{q}^4+\frac{1}{\mathrm{1\hspace{0.33em}47456}}{q}^5+\frac{37}{\mathrm{8918\hspace{0.33em}13888}}{q}^6+\mathrm{\cdots },$
ⓘ Symbols: $a_n\left(q\right)$: 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		$b_5\left(q\right)$	$=25+\frac{1}{48}{q}^2+\frac{11}{\mathrm{7\hspace{0.33em}74144}}{q}^4-\frac{1}{\mathrm{1\hspace{0.33em}47456}}{q}^5+\frac{37}{\mathrm{8918\hspace{0.33em}13888}}{q}^6+\mathrm{\cdots },$
ⓘ Symbols: $b_n\left(q\right)$: 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		$a_6\left(q\right)$	$=36+\frac{1}{70}{q}^2+\frac{187}{\mathrm{439\hspace{0.33em}04000}}{q}^4+\frac{\mathrm{67\hspace{0.33em}43617}}{\mathrm{9293\hspace{0.33em}59872\hspace{0.33em}00000}}{q}^6+\mathrm{\cdots },$
ⓘ Symbols: $a_n\left(q\right)$: 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		$b_6\left(q\right)$	$=36+\frac{1}{70}{q}^2+\frac{187}{\mathrm{439\hspace{0.33em}04000}}{q}^4-\frac{\mathrm{58\hspace{0.33em}61633}}{\mathrm{9293\hspace{0.33em}59872\hspace{0.33em}00000}}{q}^6+\mathrm{\cdots }.$
ⓘ Symbols: $b_n\left(q\right)$: 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 $m=7,8,9,\mathrm{\dots }$ are:

28.6.14		$$\begin{array}{c}\hfill a_m\left(q\right)\hfill \\ \hfill b_m\left(q\right)\hfill \end{array}\}=m^2+\frac{1}{2(m^2-1)}{q}^2+\frac{5m^2+7}{32{(m^2-1)}^3(m^2-4)}{q}^4+\frac{9m^4+58m^2+29}{64{(m^2-1)}^5(m^2-4)(m^2-9)}{q}^6+\mathrm{\cdots }.$$
ⓘ Symbols: $a_n\left(q\right)$: eigenvalues of Mathieu equation, $b_n\left(q\right)$: 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), §28.6 and Ch.28

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		$$a_m\left(q\right)-b_m\left(q\right)=\frac{2q^m}{{\left(2^{m-1}(m-1)!\right)}^2}\left(1+O\left(q^2\right)\right).$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $a_n\left(q\right)$: eigenvalues of Mathieu equation, $b_n\left(q\right)$: eigenvalues of Mathieu equation, $!$: factorial (as in $n!$), $m$: integer and $q=h^2$: parameter Permalink: http://dlmf.nist.gov/28.6.E15 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28

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

28.6.16		$$a-{(2n)}^2-\frac{q^2}{a-{(2n-2)}^2-}\frac{q^2}{a-{(2n-4)}^2-}\mathrm{\cdots }\frac{q^2}{a-2^2-}\frac{2q^2}{a}=-\frac{q^2}{{(2n+2)}^2-a-}\frac{q^2}{{(2n+4)}^2-a-}\mathrm{\cdots },$$
$a=a_{2n}\left(q\right)$,
ⓘ Symbols: $a_n\left(q\right)$: 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		$$a-{(2n+1)}^2-\frac{q^2}{a-{(2n-1)}^2-}\mathrm{\cdots }\frac{q^2}{a-3^2-}\frac{q^2}{a-1^2-q}=-\frac{q^2}{{(2n+3)}^2-a-}\frac{q^2}{{(2n+5)}^2-a-}\mathrm{\cdots },$$
$a=a_{2n+1}\left(q\right)$,
ⓘ Symbols: $a_n\left(q\right)$: 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		$$a-{(2n+1)}^2-\frac{q^2}{a-{(2n-1)}^2-}\mathrm{\cdots }\frac{q^2}{a-3^2-}\frac{q^2}{a-1^2+q}=-\frac{q^2}{{(2n+3)}^2-a-}\frac{q^2}{{(2n+5)}^2-a-}\mathrm{\cdots },$$
$a=b_{2n+1}\left(q\right)$,
ⓘ Symbols: $b_n\left(q\right)$: 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		$$a-{(2n+2)}^2-\frac{q^2}{a-{(2n)}^2-}\frac{q^2}{a-{(2n-2)}^2-}\mathrm{\cdots }\frac{q^2}{a-2^2}=-\frac{q^2}{{(2n+4)}^2-a-}\frac{q^2}{{(2n+6)}^2-a-}\mathrm{\cdots },$$
$a=b_{2n+2}\left(q\right)$.
ⓘ Symbols: $b_n\left(q\right)$: 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 $n=0,1,\mathrm{\dots },9$ 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		$$\underset{n\to \mathrm{\infty }}{lim\; inf}\frac{{\rho }_n^{(j)}}{n^2}\ge k{k}^{\prime }{(K\left(k\right))}^2=\mathrm{2.04183\hspace{0.33em}4}\mathrm{\dots },$$
ⓘ Symbols: $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $lim\; inf$: least limit point, $n$: integer, $j$: integer, ${\rho }_n^{(j)}$: radii of convergence and $k$: root Referenced by: §28.6(i), §28.6(ii) Permalink: http://dlmf.nist.gov/28.6.E20 Encodings: TeX, pMML, png See also: Annotations for §28.6(i), §28.6 and Ch.28

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).

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

28.6.21		$2^{1/2}{\mathrm{ce}}_0(z,q)$	$=1-\frac{1}{2}q\cos 2z+\frac{1}{32}{q}^2\left(\cos 4z-2\right)-\frac{1}{128}{q}^3\left(\frac{1}{9}\cos 6z-11\cos 2z\right)+\mathrm{\cdots },$
ⓘ Symbols: ${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: 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		${\mathrm{ce}}_1(z,q)$	$=\begin{array}{l}\begin{array}{l}\cos z-\frac{1}{8}q\cos 3z+\frac{1}{128}{q}^2\left(\frac{2}{3}\cos 5z-2\cos 3z-\cos z\right)\\ -\frac{1}{1024}{q}^3\left(\frac{1}{9}\cos 7z-\frac{8}{9}\cos 5z-\frac{1}{3}\cos 3z+2\cos z\right)\end{array}\\ +\mathrm{\cdots },\end{array}$
ⓘ Symbols: ${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $q=h^2$: parameter and $z$: complex variable Permalink: http://dlmf.nist.gov/28.6.E22 Encodings: TeX, pMML, png See also: Annotations for §28.6(ii), §28.6 and Ch.28
28.6.23		${\mathrm{se}}_1(z,q)$	$=\begin{array}{l}\begin{array}{l}\sin z-\frac{1}{8}q\sin 3z+\frac{1}{128}{q}^2\left(\frac{2}{3}\sin 5z+2\sin 3z-\sin z\right)\\ -\frac{1}{1024}{q}^3\left(\frac{1}{9}\sin 7z+\frac{8}{9}\sin 5z-\frac{1}{3}\sin 3z-2\sin z\right)\end{array}\\ +\mathrm{\cdots },\end{array}$
ⓘ Symbols: ${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $q=h^2$: parameter and $z$: complex variable Permalink: http://dlmf.nist.gov/28.6.E23 Encodings: TeX, pMML, png See also: Annotations for §28.6(ii), §28.6 and Ch.28
28.6.24		${\mathrm{ce}}_2(z,q)$	$=\cos 2z-\frac{1}{4}q\left(\frac{1}{3}\cos 4z-1\right)+\frac{1}{128}{q}^2\left(\frac{1}{3}\cos 6z-\frac{76}{9}\cos 2z\right)+\mathrm{\cdots },$
ⓘ Symbols: ${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $q=h^2$: parameter and $z$: complex variable Permalink: http://dlmf.nist.gov/28.6.E24 Encodings: TeX, pMML, png See also: Annotations for §28.6(ii), §28.6 and Ch.28
28.6.25		${\mathrm{se}}_2(z,q)$	$=\sin 2z-\frac{1}{12}q\sin 4z+\frac{1}{128}{q}^2\left(\frac{1}{3}\sin 6z-\frac{4}{9}\sin 2z\right)+\mathrm{\cdots }.$
ⓘ Symbols: ${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $q=h^2$: parameter and $z$: complex variable Permalink: http://dlmf.nist.gov/28.6.E25 Encodings: TeX, pMML, png See also: Annotations for §28.6(ii), §28.6 and Ch.28

For $m=3,4,5,\mathrm{\dots }$,

28.6.26		$${\mathrm{ce}}_m(z,q)=\begin{array}{l}\cos mz-\frac{q}{4}\left(\frac{1}{m+1}\cos (m+2)z-\frac{1}{m-1}\cos (m-2)z\right)\\ +\frac{q^2}{32}\left(\frac{1}{(m+1)(m+2)}\cos (m+4)z+\frac{1}{(m-1)(m-2)}\cos (m-4)z-\frac{2(m^2+1)}{{(m^2-1)}^2}\cos mz\right)+\mathrm{\cdots }.\end{array}$$
ⓘ Symbols: ${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: 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), §28.6 and Ch.28

For the corresponding expansions of ${\mathrm{se}}_m(z,q)$ for $m=3,4,5,\mathrm{\dots }$ change $\cos$ to $\sin$ 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 $a_n\left(q\right)$ and $b_n\left(q\right)$; compare Table 28.6.1 and (28.6.20).