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1: 28.30 Expansions in Series of Eigenfunctions

§28.30 Expansions in Series of Eigenfunctions

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§28.30(i) Real Variable

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

2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ► … ► The analogous orthonormality is … ►and completeness relation … ► … ►

§1.18(viii) Mixed Spectra and Eigenfunction Expansions

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3: Howard S. Cohl

… ►Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and $q$-series. …

4: 31.17 Physical Applications

… ►for the common eigenfunction $\mathrm{\Psi }(𝐱)=\mathrm{\Psi }(x_s,x_t,x_u)$, where $a$ is the coupling parameter of interacting spins. …The operators ${𝐉}^2$ and ${𝐻}_s$ admit separation of variables in $z_1,z_2$, leading to the following factorization of the eigenfunction $\mathrm{\Psi }(𝐱)$: ►

31.17.4 $$\mathrm{\Psi }(𝐱)={(z_1{z}_2)}^{-s-\frac{1}{4}}{((z_1-1)(z_2-1))}^{-t-\frac{1}{4}}{((z_1-a)(z_2-a))}^{-u-\frac{1}{4}}w(z_1)w(z_2),$$

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Symbols:

$z$: complex variable, $a$: complex parameter and $\mathrm{\Psi }(𝐱)$: common eigenfunction

Permalink:

http://dlmf.nist.gov/31.17.E4

Encodings:

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See also:

Annotations for §31.17(i), §31.17 and Ch.31

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5: 18.39 Physical Applications

… ►The corresponding eigenfunctions are … ►The eigenfunctions of one of the separated ordinary differential equations are Legendre polynomials. … ►For a third example, one in which the eigenfunctions are Laguerre polynomials, see Seaborn (1991, pp. 87-93) and Nikiforov and Uvarov (1988, pp. 76-80 and 320-323). …

6: 28.12 Definitions and Basic Properties

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§28.12(ii) Eigenfunctions ${\mathrm{me}}_{\nu }(z,q)$

►Two eigenfunctions correspond to each eigenvalue $a={\lambda }_{\nu }\left(q\right)$. …The other eigenfunction is ${\mathrm{me}}_{\nu }(-z,q)$, a Floquet solution with respect to $-\nu$ with $a={\lambda }_{\nu }\left(q\right)$. …

7: 28.2 Definitions and Basic Properties

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§28.2(vi) Eigenfunctions

►Table 28.2.2 gives the notation for the eigenfunctions corresponding to the eigenvalues in Table 28.2.1. Period $\pi$ means that the eigenfunction has the property $w(z+\pi )=w(z)$, whereas antiperiod $\pi$ means that $w(z+\pi )=-w(z)$. … ►

Table 28.2.2: Eigenfunctions of Mathieu’s equation.

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Eigenvalues	Eigenfunctions	Periodicity	Parity
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8: 29.3 Definitions and Basic Properties

… ►The eigenfunctions corresponding to the eigenvalues of §29.3(i) are denoted by ${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)$, ${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)$. … ►

Table 29.3.2: Lamé functions.

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boundary conditions	eigenvalue $h$	eigenfunction $w(z)$	parity of $w(z)$	parity of $w(z-K)$	period of $w(z)$
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9: 30.9 Asymptotic Approximations and Expansions

… ►For the eigenfunctions see Meixner and Schäfke (1954, §3.251) and Müller (1963). … ►For the eigenfunctions see Meixner and Schäfke (1954, §3.252) and Müller (1962). …

10: 3.7 Ordinary Differential Equations

… ►The values ${\lambda }_k$ are the eigenvalues and the corresponding solutions $w_k$ of the differential equation are the eigenfunctions. The eigenvalues ${\lambda }_k$ are simple, that is, there is only one corresponding eigenfunction (apart from a normalization factor), and when ordered increasingly the eigenvalues satisfy …