28 Mathieu Functions and Hill’s EquationMathieu Functions of Noninteger Order28.14 Fourier Series28.16 Asymptotic Expansions for Large $q$

§28.15 Expansions for Small $q$

ⓘ

Keywords:

Fourier coefficients, Mathieu functions, asymptotic forms for small $q$

Permalink:

http://dlmf.nist.gov/28.15

See also:

Annotations for Ch.28

Contents

1. §28.15(i) Eigenvalues ${\lambda }_{\nu }\left(q\right)$
2. §28.15(ii) Solutions ${\mathrm{me}}_{\nu }(z,q)$

§28.15(i) Eigenvalues ${\lambda }_{\nu }\left(q\right)$

ⓘ

Keywords:

Mathieu’s equation, continued-fraction equations, eigenvalues (or characteristic values), power-series expansions in $q$

Notes:

See Meixner and Schäfke (1954, pp. 119 and 117).

Referenced by:

item (a)

Permalink:

http://dlmf.nist.gov/28.15.i

See also:

Annotations for §28.15 and Ch.28

28.15.1		$${\lambda }_{\nu }\left(q\right)={\nu }^2+\frac{1}{2({\nu }^2-1)}{q}^2+\frac{5{\nu }^2+7}{32{({\nu }^2-1)}^3({\nu }^2-4)}{q}^4+\frac{9{\nu }^4+58{\nu }^2+29}{64{({\nu }^2-1)}^5({\nu }^2-4)({\nu }^2-9)}{q}^6+\mathrm{\cdots }.$$
ⓘ Symbols: ${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $\nu$: complex parameter A&S Ref: 20.3.15 (in slightly different form) Permalink: http://dlmf.nist.gov/28.15.E1 Encodings: TeX, pMML, png See also: Annotations for §28.15(i), §28.15 and Ch.28

Higher coefficients can be found by equating powers of $q$ in the following continued-fraction equation, with $a={\lambda }_{\nu }\left(q\right)$:

28.15.2		$$a-{\nu }^2-\frac{q^2}{a-{(\nu +2)}^2-}\frac{q^2}{a-{(\nu +4)}^2-}\mathrm{\cdots }=\frac{q^2}{a-{(\nu -2)}^2-}\frac{q^2}{a-{(\nu -4)}^2-}\mathrm{\cdots }.$$
ⓘ Symbols: $q=h^2$: parameter, $\nu$: complex parameter and $a$: parameter Referenced by: item (e) Permalink: http://dlmf.nist.gov/28.15.E2 Encodings: TeX, pMML, png See also: Annotations for §28.15(i), §28.15 and Ch.28

§28.15(ii) Solutions ${\mathrm{me}}_{\nu }(z,q)$

ⓘ

Keywords:

Mathieu functions, power series in $q$

Notes:

See Meixner and Schäfke (1954, p. 122).

Referenced by:

item (a)

Permalink:

http://dlmf.nist.gov/28.15.ii

See also:

Annotations for §28.15 and Ch.28

28.15.3		$${\mathrm{me}}_{\nu }(z,q)=\begin{array}{l}{\mathrm{e}}^{\mathrm{i}\nu z}-\frac{q}{4}\left(\frac{1}{\nu +1}{\mathrm{e}}^{\mathrm{i}(\nu +2)z}-\frac{1}{\nu -1}{\mathrm{e}}^{\mathrm{i}(\nu -2)z}\right)\\ +\frac{q^2}{32}\left(\frac{1}{(\nu +1)(\nu +2)}{\mathrm{e}}^{\mathrm{i}(\nu +4)z}+\frac{1}{(\nu -1)(\nu -2)}{\mathrm{e}}^{\mathrm{i}(\nu -4)z}-\frac{2({\nu }^2+1)}{{({\nu }^2-1)}^2}{\mathrm{e}}^{\mathrm{i}\nu z}\right)+\mathrm{\cdots };\end{array}$$
ⓘ Symbols: ${\mathrm{me}}_n(z,q)$: Mathieu function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter A&S Ref: 20.3.17 (only two terms and without normalization) Permalink: http://dlmf.nist.gov/28.15.E3 Encodings: TeX, pMML, png See also: Annotations for §28.15(ii), §28.15 and Ch.28

compare §28.6(ii).

28.14 Fourier Series28.16 Asymptotic Expansions for Large $q$

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