§28.30 Expansions in Series of Eigenfunctions

Keywords:: Hill’s equation
Referenced by:: Ch.28
Permalink:: http://dlmf.nist.gov/28.30

§28.30(i) Real Variable
§28.30(ii) Complex Variable

§28.30(i) Real Variable

Notes:: See Magnus and Winkler (1966, pp. 44).
Keywords:: Hill’s equation
Permalink:: http://dlmf.nist.gov/28.30.i

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 $w_{m}^{{\prime\prime}}+(\widehat{\lambda}_{m}+Q(x))w_{m}=0,$

Symbols:: $m$ : integer, $x$ : real variable and $w(z)$ : Mathieu’s equation solution
Permalink:: http://dlmf.nist.gov/28.30.E1
Encodings:: TeX, pMML, png

28.30.2 $\frac{1}{2\pi}\int _{0}^{{2\pi}}w_{m}(x)w_{n}(x)dx=\delta _{{m,n}}.$

Symbols:: $\delta _{{j,k}}$ : Kronecker delta, $dx$ : 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

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