28 Mathieu Functions and Hill’s Equation Hill’s Equation 28.29 Definitions and Basic Properties 28.31 Equations of Whittaker–Hill and Ince

§28.30 Expansions in Series of Eigenfunctions

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Keywords:: Hill’s equation, expansions in series of eigenfunctions
Referenced by:: Ch.28
Permalink:: http://dlmf.nist.gov/28.30
See also:: Annotations for Ch.28

§28.30(i) Real Variable

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Keywords:: Hill’s equation, eigenfunctions
Notes:: See Magnus and Winkler (1966, pp. 44).
Permalink:: http://dlmf.nist.gov/28.30.i
See also:: Annotations for §28.30 and Ch.28

Let $\widehat{\lambda}_{m}$ , $m=0,1,2,\dots$ , be the set of characteristic values (28.29.16) and (28.29.17), arranged in their natural order (see (28.29.18)), and let $w_{m}(x)$ , $m=0,1,2,\dots$ , be the eigenfunctions, that is, an orthonormal set of $2\pi$ -periodic solutions; thus

28.30.1	$\displaystyle w_{m}^{\prime\prime}+(\widehat{\lambda}_{m}+Q(x))w_{m}$	$\displaystyle=0,$
ⓘ Symbols: $m$ : integer, $x$ : real variable, $w(z)$ : Mathieu’s equation solution and $\widehat{\lambda}_{m}$ : characteristic values Permalink: http://dlmf.nist.gov/28.30.E1 Encodings: TeX, pMML, png See also: Annotations for §28.30(i), §28.30 and Ch.28
28.30.2	$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}w_{m}(x)w_{n}(x)\,\mathrm{d}x$	$\displaystyle=\delta_{m,n}.$
ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $m$ : integer, $n$ : integer, $x$ : real variable and $w(z)$ : Mathieu’s equation solution Permalink: http://dlmf.nist.gov/28.30.E2 Encodings: TeX, pMML, png See also: Annotations for §28.30(i), §28.30 and Ch.28

Then every continuous $2\pi$ -periodic function $f(x)$ whose second derivative is square-integrable over the interval $[0,2\pi]$ can be expanded in a uniformly and absolutely convergent series

28.30.3		$f(x)=\sum_{m=0}^{\infty}f_{m}w_{m}(x),$
ⓘ Symbols: $m$ : integer, $x$ : real variable and $w(z)$ : Mathieu’s equation solution Permalink: http://dlmf.nist.gov/28.30.E3 Encodings: TeX, pMML, png See also: Annotations for §28.30(i), §28.30 and Ch.28

where

28.30.4		$f_{m}=\frac{1}{2\pi}\int_{0}^{2\pi}f(x)w_{m}(x)\,\mathrm{d}x.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $m$ : integer, $x$ : real variable and $w(z)$ : Mathieu’s equation solution Permalink: http://dlmf.nist.gov/28.30.E4 Encodings: TeX, pMML, png See also: Annotations for §28.30(i), §28.30 and Ch.28

§28.30(ii) Complex Variable

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Permalink:: http://dlmf.nist.gov/28.30.ii
See also:: Annotations for §28.30 and Ch.28

For analogous results to those of §28.19, see Schäfke (1960, 1961b), and Meixner et al. (1980, §1.1.11).

28.29 Definitions and Basic Properties 28.31 Equations of Whittaker–Hill and Ince

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§28.30 Expansions in Series of Eigenfunctions

Contents

§28.30(i) Real Variable

§28.30(ii) Complex Variable