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

§28.6 Expansions for Small

Permalink:: http://dlmf.nist.gov/28.6

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

§28.6(i) Eigenvalues

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).
Keywords:: Mathieu’s equation
Referenced by:: §28.2(v), §28.34(ii)
Permalink:: http://dlmf.nist.gov/28.6.i

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

Symbols:: $\mathop{a_{{n}}\/}\nolimits\!\left(q\right)$ : 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

28.6.2 $\mathop{a_{{1}}\/}\nolimits\!\left(q\right)=1+q-\tfrac{1}{8}q^{2}-\tfrac{1}{64% }q^{3}-\tfrac{1}{1536}q^{4}+\tfrac{11}{36864}q^{5}+\tfrac{49}{5\;89824}q^{6}+% \tfrac{55}{94\;37184}q^{7}-\tfrac{83}{353\;89440}q^{8}+\cdots,$

Symbols:: $\mathop{a_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.6.E2
Encodings:: TeX, pMML, png

28.6.3 $\mathop{b_{{1}}\/}\nolimits\!\left(q\right)=1-q-\tfrac{1}{8}q^{2}+\tfrac{1}{64% }q^{3}-\tfrac{1}{1536}q^{4}-\tfrac{11}{36864}q^{5}+\tfrac{49}{5\;89824}q^{6}-% \tfrac{55}{94\;37184}q^{7}-\tfrac{83}{353\;89440}q^{8}+\cdots,$

Symbols:: $\mathop{b_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.6.E3
Encodings:: TeX, pMML, png

28.6.4 $\mathop{a_{{2}}\/}\nolimits\!\left(q\right)=4+\tfrac{5}{12}q^{2}-\tfrac{763}{1% 3824}q^{4}+\tfrac{10\;02401}{796\;26240}q^{6}-\tfrac{16690\;68401}{45\;86471\;% 42400}q^{8}+\cdots,$

Symbols:: $\mathop{a_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.6.E4
Encodings:: TeX, pMML, png

28.6.5 $\mathop{b_{{2}}\/}\nolimits\!\left(q\right)=4-\tfrac{1}{12}q^{2}+\tfrac{5}{138% 24}q^{4}-\tfrac{289}{796\;26240}q^{6}+\tfrac{21391}{45\;86471\;42400}q^{8}+\cdots,$

Symbols:: $\mathop{b_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.6.E5
Encodings:: TeX, pMML, png

28.6.6 $\mathop{a_{{3}}\/}\nolimits\!\left(q\right)=9+\tfrac{1}{16}q^{2}+\tfrac{1}{64}% q^{3}+\tfrac{13}{20480}q^{4}-\tfrac{5}{16384}q^{5}-\tfrac{1961}{235\;92960}q^{% 6}-\tfrac{609}{1048\;57600}q^{7}+\cdots,$

Symbols:: $\mathop{a_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.6.E6
Encodings:: TeX, pMML, png

28.6.7 $\mathop{b_{{3}}\/}\nolimits\!\left(q\right)=9+\tfrac{1}{16}q^{2}-\tfrac{1}{64}% q^{3}+\tfrac{13}{20480}q^{4}+\tfrac{5}{16384}q^{5}-\tfrac{1961}{235\;92960}q^{% 6}+\tfrac{609}{1048\;57600}q^{7}+\cdots,$

Symbols:: $\mathop{b_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.6.E7
Encodings:: TeX, pMML, png

28.6.8 $\mathop{a_{{4}}\/}\nolimits\!\left(q\right)=16+\tfrac{1}{30}q^{2}+\tfrac{433}{% 8\;64000}q^{4}-\tfrac{5701}{27216\;00000}q^{6}+\cdots,$

Symbols:: $\mathop{a_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.6.E8
Encodings:: TeX, pMML, png

28.6.9 $\mathop{b_{{4}}\/}\nolimits\!\left(q\right)=16+\tfrac{1}{30}q^{2}-\tfrac{317}{% 8\;64000}q^{4}+\tfrac{10049}{27216\;00000}q^{6}+\cdots,$

Symbols:: $\mathop{b_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.6.E9
Encodings:: TeX, pMML, png

28.6.10 $\mathop{a_{{5}}\/}\nolimits\!\left(q\right)=25+\tfrac{1}{48}q^{2}+\tfrac{11}{7% \;74144}q^{4}+\tfrac{1}{1\;47456}q^{5}+\tfrac{37}{8918\;13888}q^{6}+\cdots,$

Symbols:: $\mathop{a_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.6.E10
Encodings:: TeX, pMML, png

28.6.11 $\mathop{b_{{5}}\/}\nolimits\!\left(q\right)=25+\tfrac{1}{48}q^{2}+\tfrac{11}{7% \;74144}q^{4}-\tfrac{1}{1\;47456}q^{5}+\tfrac{37}{8918\;13888}q^{6}+\cdots,$

Symbols:: $\mathop{b_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.6.E11
Encodings:: TeX, pMML, png

28.6.12 $\mathop{a_{{6}}\/}\nolimits\!\left(q\right)=36+\tfrac{1}{70}q^{2}+\tfrac{187}{% 439\;04000}q^{4}+\tfrac{67\;43617}{9293\;59872\;00000}q^{6}+\cdots,$

Symbols:: $\mathop{a_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.6.E12
Encodings:: TeX, pMML, png

28.6.13 $\mathop{b_{{6}}\/}\nolimits\!\left(q\right)=36+\tfrac{1}{70}q^{2}+\tfrac{187}{% 439\;04000}q^{4}-\tfrac{58\;61633}{9293\;59872\;00000}q^{6}+\cdots.$

Symbols:: $\mathop{b_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.6.E13
Encodings:: TeX, pMML, png

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:: $\mathop{a_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, $\mathop{b_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, : 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

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{a_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, $\mathop{b_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, : : factorial, : integer and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.6.E15
Encodings:: TeX, pMML, png

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:: $\mathop{a_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter, : integer and : parameter
Referenced by:: §28.34(ii)
Permalink:: http://dlmf.nist.gov/28.6.E16
Encodings:: TeX, pMML, png

28.6.17 $a-(2n+1)^{2}-\cfrac{q^{2}}{a-(2n-1)^{2}-}\cdots\cfrac{q^{2}}{a-3^{2}-\cfrac{q^% {2}}{a-1^{2}-q}}=-\cfrac{q^{2}}{(2n+3)^{2}-a-\cfrac{q^{2}}{(2n+5)^{2}-a-\cdots% }},$ $a=\mathop{a_{{2n+1}}\/}\nolimits\!\left(q\right)$ ,

Symbols:: $\mathop{a_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter, : integer and : parameter
Permalink:: http://dlmf.nist.gov/28.6.E17
Encodings:: TeX, pMML, png

28.6.18 $a-(2n+1)^{2}-\cfrac{q^{2}}{a-(2n-1)^{2}-}\cdots\cfrac{q^{2}}{a-3^{2}-\cfrac{q^% {2}}{a-1^{2}+q}}=-\cfrac{q^{2}}{(2n+3)^{2}-a-\cfrac{q^{2}}{(2n+5)^{2}-a-\cdots% }},$ $a=\mathop{b_{{2n+1}}\/}\nolimits\!\left(q\right)$ ,

Symbols:: $\mathop{b_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter, : integer and : parameter
Permalink:: http://dlmf.nist.gov/28.6.E18
Encodings:: TeX, pMML, png

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:: $\mathop{b_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter, : integer and : parameter
Referenced by:: §28.34(ii)
Permalink:: http://dlmf.nist.gov/28.6.E19
Encodings:: TeX, pMML, png

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 $\mathop{a_{{2n}}\/}\nolimits\!\left(q\right)$ , for $\mathop{b_{{2n+2}}\/}\nolimits\!\left(q\right)$ , and 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).)

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

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

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

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : 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

where is the unique root of the equation $2\!\mathop{E\/}\nolimits\!\left(k\right)=\mathop{K\/}\nolimits\!\left(k\right)$ in the interval , 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 $\mathop{\mathrm{se}_{{n}}\/}\nolimits$

Notes:: See Meixner and Schäfke (1954, pp. 122–123).
Keywords:: Mathieu functions
Referenced by:: §28.15(ii), §28.34(iii)
Permalink:: http://dlmf.nist.gov/28.6.ii

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

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

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : 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

28.6.22 $\mathop{\mathrm{ce}_{{1}}\/}\nolimits\!\left(z,q\right)=\mathop{\cos\/}% \nolimits z-\tfrac{1}{8}q\mathop{\cos\/}\nolimits 3z+\tfrac{1}{128}q^{2}\left(% \tfrac{2}{3}\mathop{\cos\/}\nolimits 5z-2\mathop{\cos\/}\nolimits 3z-\mathop{% \cos\/}\nolimits z\right)-\tfrac{1}{1024}q^{3}\left(\tfrac{1}{9}\mathop{\cos\/% }\nolimits 7z-\tfrac{8}{9}\mathop{\cos\/}\nolimits 5z-\tfrac{1}{3}\mathop{\cos% \/}\nolimits 3z+2\mathop{\cos\/}\nolimits z\right)+\cdots,$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : cosine function, $q=h^{2}$ : parameter and : complex variable
Permalink:: http://dlmf.nist.gov/28.6.E22
Encodings:: TeX, pMML, png

28.6.23 $\mathop{\mathrm{se}_{{1}}\/}\nolimits\!\left(z,q\right)=\mathop{\sin\/}% \nolimits z-\tfrac{1}{8}q\mathop{\sin\/}\nolimits 3z+\tfrac{1}{128}q^{2}\left(% \tfrac{2}{3}\mathop{\sin\/}\nolimits 5z+2\mathop{\sin\/}\nolimits 3z-\mathop{% \sin\/}\nolimits z\right)-\tfrac{1}{1024}q^{3}\left(\tfrac{1}{9}\mathop{\sin\/% }\nolimits 7z+\tfrac{8}{9}\mathop{\sin\/}\nolimits 5z-\tfrac{1}{3}\mathop{\sin% \/}\nolimits 3z-2\mathop{\sin\/}\nolimits z\right)+\cdots,$

Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\sin\/}\nolimits z$ : sine function, $q=h^{2}$ : parameter and : complex variable
Permalink:: http://dlmf.nist.gov/28.6.E23
Encodings:: TeX, pMML, png

28.6.24 $\mathop{\mathrm{ce}_{{2}}\/}\nolimits\!\left(z,q\right)=\mathop{\cos\/}% \nolimits 2z-\tfrac{1}{4}q\left(\tfrac{1}{3}\mathop{\cos\/}\nolimits 4z-1% \right)+\tfrac{1}{128}q^{2}\left(\tfrac{1}{3}\mathop{\cos\/}\nolimits 6z-% \tfrac{76}{9}\mathop{\cos\/}\nolimits 2z\right)+\cdots,$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : cosine function, $q=h^{2}$ : parameter and : complex variable
Permalink:: http://dlmf.nist.gov/28.6.E24
Encodings:: TeX, pMML, png

28.6.25 $\mathop{\mathrm{se}_{{2}}\/}\nolimits\!\left(z,q\right)=\mathop{\sin\/}% \nolimits 2z-\tfrac{1}{12}q\mathop{\sin\/}\nolimits 4z+\tfrac{1}{128}q^{2}% \left(\tfrac{1}{3}\mathop{\sin\/}\nolimits 6z-\tfrac{4}{9}\mathop{\sin\/}% \nolimits 2z\right)+\cdots.$

Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\sin\/}\nolimits z$ : sine function, $q=h^{2}$ : parameter and : complex variable
Permalink:: http://dlmf.nist.gov/28.6.E25
Encodings:: TeX, pMML, png

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:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : 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)
Permalink:: http://dlmf.nist.gov/28.6.E26
Encodings:: TeX, pMML, png

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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 28.5 Second Solutions $\mathop{\mathrm{fe}_{{n}}\/}\nolimits$ , $\mathop{\mathrm{ge}_{{n}}\/}\nolimits$ 28.7 Analytic Continuation of Eigenvalues

§28.6 Expansions for Small

Contents

§28.6(i) Eigenvalues

§28.6(ii) Functions and

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