28 Mathieu Functions and Hill’s Equation Mathieu Functions of Integer Order 28.5 Second Solutions $\operatorname{fe}_{n}$ , $\operatorname{ge}_{n}$ 28.7 Analytic Continuation of Eigenvalues

§28.6 Expansions for Small

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Permalink:: http://dlmf.nist.gov/28.6
See also:: Annotations for Ch.28

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

§28.6(i) Eigenvalues

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Keywords:: Mathieu’s equation, continued-fraction equations, eigenvalues (or characteristic values), power-series expansions in
Notes:: See Meixner and Schäfke (1954, pp. 118, 120, and 122) and Meixner et al. (1980, p. 101 and §2.4). For (28.6.20) see Volkmer (1998).
Referenced by:: §28.2(v), item (a), 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.i
Clarification (effective with 1.1.10):: Just below (28.6.14), a sentence was added referring the reader to Frenkel and Portugal (2001, §2).
See also:: Annotations for §28.6 and Ch.28

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 $a_{0}\left(q\right)=-\tfrac{1}{2}q^{2}+\tfrac{7}{128}q^{4}-\tfrac{29}{2304}q^{% 6}+\tfrac{68687}{188\;74368}q^{8}+\cdots,$

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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 $a_{1}\left(q\right)=1+q-\tfrac{1}{8}q^{2}-\tfrac{1}{64}q^{3}-\tfrac{1}{1536}q^% {4}+\tfrac{11}{36864}q^{5}+\tfrac{49}{5\;89824}q^{6}+\tfrac{55}{94\;37184}q^{7% }-\tfrac{83}{353\;89440}q^{8}+\cdots,$

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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 $b_{1}\left(q\right)=1-q-\tfrac{1}{8}q^{2}+\tfrac{1}{64}q^{3}-\tfrac{1}{1536}q^% {4}-\tfrac{11}{36864}q^{5}+\tfrac{49}{5\;89824}q^{6}-\tfrac{55}{94\;37184}q^{7% }-\tfrac{83}{353\;89440}q^{8}+\cdots,$

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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 $a_{2}\left(q\right)=4+\tfrac{5}{12}q^{2}-\tfrac{763}{13824}q^{4}+\tfrac{10\;02% 401}{796\;26240}q^{6}-\tfrac{16690\;68401}{45\;86471\;42400}q^{8}+\cdots,$

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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 $b_{2}\left(q\right)=4-\tfrac{1}{12}q^{2}+\tfrac{5}{13824}q^{4}-\tfrac{289}{796% \;26240}q^{6}+\tfrac{21391}{45\;86471\;42400}q^{8}+\cdots,$

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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 $a_{3}\left(q\right)=9+\tfrac{1}{16}q^{2}+\tfrac{1}{64}q^{3}+\tfrac{13}{20480}q% ^{4}-\tfrac{5}{16384}q^{5}-\tfrac{1961}{235\;92960}q^{6}-\tfrac{609}{1048\;576% 00}q^{7}+\cdots,$

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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 $b_{3}\left(q\right)=9+\tfrac{1}{16}q^{2}-\tfrac{1}{64}q^{3}+\tfrac{13}{20480}q% ^{4}+\tfrac{5}{16384}q^{5}-\tfrac{1961}{235\;92960}q^{6}+\tfrac{609}{1048\;576% 00}q^{7}+\cdots,$

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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 $a_{4}\left(q\right)=16+\tfrac{1}{30}q^{2}+\tfrac{433}{8\;64000}q^{4}-\tfrac{57% 01}{27216\;00000}q^{6}+\cdots,$

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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 $b_{4}\left(q\right)=16+\tfrac{1}{30}q^{2}-\tfrac{317}{8\;64000}q^{4}+\tfrac{10% 049}{27216\;00000}q^{6}+\cdots,$

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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 $a_{5}\left(q\right)=25+\tfrac{1}{48}q^{2}+\tfrac{11}{7\;74144}q^{4}+\tfrac{1}{% 1\;47456}q^{5}+\tfrac{37}{8918\;13888}q^{6}+\cdots,$

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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 $b_{5}\left(q\right)=25+\tfrac{1}{48}q^{2}+\tfrac{11}{7\;74144}q^{4}-\tfrac{1}{% 1\;47456}q^{5}+\tfrac{37}{8918\;13888}q^{6}+\cdots,$

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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 $a_{6}\left(q\right)=36+\tfrac{1}{70}q^{2}+\tfrac{187}{439\;04000}q^{4}+\tfrac{% 67\;43617}{9293\;59872\;00000}q^{6}+\cdots,$

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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 $b_{6}\left(q\right)=36+\tfrac{1}{70}q^{2}+\tfrac{187}{439\;04000}q^{4}-\tfrac{% 58\;61633}{9293\;59872\;00000}q^{6}+\cdots.$

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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 $\rselection{a_{m}\left(q\right)\\ b_{m}\left(q\right)}=m^{2}+\frac{1}{2(m^{2}-1)}q^{2}+\frac{5m^{2}+7}{32(m^{2}-% 1)^{3}(m^{2}-4)}q^{4}+\frac{9m^{4}+58m^{2}+29}{64(m^{2}-1)^{5}(m^{2}-4)(m^{2}-% 9)}q^{6}+\cdots.$

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Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, : 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 $a_{m}\left(q\right)-b_{m}\left(q\right)=\frac{2q^{m}}{\left(2^{m-1}(m-1)!% \right)^{2}}\left(1+O\left(q^{2}\right)\right).$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, : factorial (as in ), : 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 $a-(2n)^{2}-\cfrac{q^{2}}{a-(2n-2)^{2}-\cfrac{q^{2}}{a-(2n-4)^{2}-}}\cdots% \cfrac{q^{2}}{a-2^{2}-\cfrac{2q^{2}}{a}}=-\cfrac{q^{2}}{(2n+2)^{2}-a-\cfrac{q^% {2}}{(2n+4)^{2}-a-\cdots}},$ $a=a_{2n}\left(q\right)$ ,

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Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter, : integer and : 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 $a-(2n+1)^{2}-\cfrac{q^{2}}{a-(2n-1)^{2}-}\cdots\cfrac{q^{2}}{a-3^{2}-\cfrac{q^% {2}}{a-1^{2}-q}}=-\cfrac{q^{2}}{(2n+3)^{2}-a-\cfrac{q^{2}}{(2n+5)^{2}-a-\cdots% }},$ $a=a_{2n+1}\left(q\right)$ ,

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Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter, : integer and : 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 $a-(2n+1)^{2}-\cfrac{q^{2}}{a-(2n-1)^{2}-}\cdots\cfrac{q^{2}}{a-3^{2}-\cfrac{q^% {2}}{a-1^{2}+q}}=-\cfrac{q^{2}}{(2n+3)^{2}-a-\cfrac{q^{2}}{(2n+5)^{2}-a-\cdots% }},$ $a=b_{2n+1}\left(q\right)$ ,

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Symbols:: $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter, : integer and : 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 $a-(2n+2)^{2}-\cfrac{q^{2}}{a-(2n)^{2}-\cfrac{q^{2}}{a-(2n-2)^{2}-}}\cdots% \cfrac{q^{2}}{a-2^{2}}=-\cfrac{q^{2}}{(2n+4)^{2}-a-\cfrac{q^{2}}{(2n+6)^{2}-a-% \cdots}},$ $a=b_{2n+2}\left(q\right)$ .

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Symbols:: $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter, : integer and : 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 for $a_{2n}\left(q\right)$ , for $b_{2n+2}\left(q\right)$ , and 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).)

Table 28.6.1: Radii of convergence for power-series expansions of eigenvalues of Mathieu’s equation.

	$\rho^{(1)}_{n}$	$\rho^{(2)}_{n}$	$\rho^{(3)}_{n}$
0 or 1	1.46876 86138	6.92895 47588	3.76995 74940
2	7.26814 68935	16.80308 98254	11.27098 52655
3	16.47116 58923	30.09677 28376	22.85524 71216
4	30.42738 20960	48.13638 18593	38.52292 50099
5	47.80596 57026	69.59879 32769	58.27413 84472
6	69.92930 51764	95.80595 67052	82.10894 36067
7	95.47527 27072	125.43541 1314	110.02736 9210
8	125.76627 89677	159.81025 4642	142.02943 1279
9	159.47921 26694	197.60667 8692	178.11513 940

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Symbols:: : integer and $\rho_{n}^{(j)}$ : radii of convergence
Referenced by:: §28.6(i), §28.6(ii)
Permalink:: http://dlmf.nist.gov/28.6.T1
See also:: Annotations for §28.6(i), §28.6 and Ch.28

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

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

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Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\liminf$ : least limit point, : integer, : integer, $\rho_{n}^{(j)}$ : radii of convergence and : 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 is the unique root of the equation $2E\left(k\right)=K\left(k\right)$ in the interval , 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 $\operatorname{se}_{n}$

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Keywords:: Mathieu functions, power series in
Notes:: See Meixner and Schäfke (1954, pp. 122–123).
Referenced by:: §28.15(ii), item (a), 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.ii
Clarification (effective with 1.1.10):: Just below (28.6.26), a sentence was added referring the reader to Frenkel and Portugal (2001, §2).
See also:: Annotations for §28.6 and Ch.28

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

28.6.21 $2^{\ifrac{1}{2}}\operatorname{ce}_{0}\left(z,q\right)=1-\tfrac{1}{2}q\cos 2z+% \tfrac{1}{32}q^{2}\left(\cos 4z-2\right)-\tfrac{1}{128}q^{3}\left(\tfrac{1}{9}% \cos 6z-11\cos 2z\right)+\cdots,$

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cos\NVar{z}$ : cosine function, $q=h^{2}$ : parameter and : 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 $\operatorname{ce}_{1}\left(z,q\right)=\cos z-\tfrac{1}{8}q\cos 3z+\tfrac{1}{12% 8}q^{2}\left(\tfrac{2}{3}\cos 5z-2\cos 3z-\cos z\right)-\tfrac{1}{1024}q^{3}% \left(\tfrac{1}{9}\cos 7z-\tfrac{8}{9}\cos 5z-\tfrac{1}{3}\cos 3z+2\cos z% \right)+\cdots,$

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cos\NVar{z}$ : cosine function, $q=h^{2}$ : parameter and : 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 $\operatorname{se}_{1}\left(z,q\right)=\sin z-\tfrac{1}{8}q\sin 3z+\tfrac{1}{12% 8}q^{2}\left(\tfrac{2}{3}\sin 5z+2\sin 3z-\sin z\right)-\tfrac{1}{1024}q^{3}% \left(\tfrac{1}{9}\sin 7z+\tfrac{8}{9}\sin 5z-\tfrac{1}{3}\sin 3z-2\sin z% \right)+\cdots,$

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Symbols:: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\sin\NVar{z}$ : sine function, $q=h^{2}$ : parameter and : 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 $\operatorname{ce}_{2}\left(z,q\right)=\cos 2z-\tfrac{1}{4}q\left(\tfrac{1}{3}% \cos 4z-1\right)+\tfrac{1}{128}q^{2}\left(\tfrac{1}{3}\cos 6z-\tfrac{76}{9}% \cos 2z\right)+\cdots,$

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cos\NVar{z}$ : cosine function, $q=h^{2}$ : parameter and : 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 $\operatorname{se}_{2}\left(z,q\right)=\sin 2z-\tfrac{1}{12}q\sin 4z+\tfrac{1}{% 128}q^{2}\left(\tfrac{1}{3}\sin 6z-\tfrac{4}{9}\sin 2z\right)+\cdots.$

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Symbols:: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\sin\NVar{z}$ : sine function, $q=h^{2}$ : parameter and : 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 $m=3,4,5,\dots$ ,

28.6.26 $\operatorname{ce}_{m}\left(z,q\right)=\cos mz-\frac{q}{4}\left(\frac{1}{m+1}% \cos(m+2)z-\frac{1}{m-1}\cos(m-2)z\right)+\frac{q^{2}}{32}\left(\frac{1}{(m+1)% (m+2)}\cos(m+4)z+\frac{1}{(m-1)(m-2)}\cos(m-4)z-\frac{2(m^{2}+1)}{(m^{2}-1)^{2% }}\cos mz\right)+\cdots.$

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cos\NVar{z}$ : cosine function, : integer, $q=h^{2}$ : parameter and : 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).

28.5 Second Solutions $\operatorname{fe}_{n}$ , $\operatorname{ge}_{n}$ 28.7 Analytic Continuation of Eigenvalues

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§28.6 Expansions for Small

Contents

§28.6(i) Eigenvalues

§28.6(ii) Functions and

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