eigenfunction expansion

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11: 18.3 Definitions

§18.3 Definitions

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As eigenfunctions of second order differential operators (Bochner’s theorem, Bochner (1929)). See the differential equations $A(x)p_n^{\prime \prime }(x)+B(x)p_n^{\prime }(x)+{\lambda }_n{p}_n(x)=0$, in Table 18.8.1.

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12: Bibliography V

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On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.

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FullText, MathReview, ZentralBlatt

Referenced by:

§19.12, §19.5.

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Expansion of vacuum magnetic fields in toroidal harmonics. Comput. Phys. Comm. 81 (1-2), pp. 74–90.

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Document

Referenced by:

§14.31(i).

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Symbolic evaluation of coefficients in Airy-type asymptotic expansions. J. Math. Anal. Appl. 269 (1), pp. 317–331.

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ISSN 0022-247X, Document, MathReview, ZentralBlatt

Referenced by:

§2.4(v).

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Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.

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ISSN 0176-4276, Document, MathReview (Luoqing Li), ZentralBlatt

Referenced by:

§31.15(iii).

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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

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ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

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13: 3.7 Ordinary Differential Equations

… ►For classification of singularities of (3.7.1) and expansions of solutions in the neighborhoods of singularities, see §2.7. … ►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 … ►The method consists of a set of rules each of which is equivalent to a truncated Taylor-series expansion, but the rules avoid the need for analytic differentiations of the differential equation. …

14: Bibliography R

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High precision Chebyshev expansions for Airy functions and their derivatives. Technical report University of Birmingham Computer Centre.

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3rd item.

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On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators. Z. Angew. Math. Mech. 78 (1), pp. 66–72.

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ISSN 0044-2267, Document, MathReview Entry, ZentralBlatt

Referenced by:

§29.20(ii).

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Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.

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ISSN 0022-2488, Document, MathReview (Wolfgang zu Castell), ZentralBlatt

Referenced by:

§10.23(ii).

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15: 30.4 Functions of the First Kind

… ►The eigenfunctions of (30.2.1) that correspond to the eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$ are denoted by ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$, $n=m,m+1,m+2,\mathrm{\dots }$. … ►

§30.4(iii) Power-Series Expansion

… ►The expansion (30.4.7) converges in the norm of $L^2(-1,1)$, that is, …It is also equiconvergent with its expansion in Ferrers functions (as in (30.4.2)), that is, the difference of corresponding partial sums converges to 0 uniformly for $-1\le x\le 1$. …

16: Bibliography K

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Hypergeometric expansions of Heun polynomials. SIAM J. Math. Anal. 22 (5), pp. 1450–1459.

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ISSN 0036-1410, Document, MathReview (Peter A. McCoy), ZentralBlatt

Referenced by:

§31.16(ii).

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Addendum: “Hypergeometric expansions of Heun polynomials”. SIAM J. Math. Anal. 22 (6), pp. 1803.

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ISSN 0036-1410, Document, MathReview (Peter A. McCoy), ZentralBlatt

Referenced by:

§31.16(ii).

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Series expansions for the third incomplete elliptic integral via partial fraction decompositions. J. Comput. Appl. Math. 207 (2), pp. 331–337.

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ISSN 0377-0427, Document, MathReview (José Luis López), ZentralBlatt

Referenced by:

§19.12, §19.5.

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Asymptotic expansions of certain $q$-series and a formula of Ramanujan for specific values of the Riemann zeta function. Acta Arith. 107 (3), pp. 269–298.

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ISSN 0065-1036, Document, Link, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§17.2(i).

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Construction of differential operators having Bochner-Krall orthogonal polynomials as eigenfunctions. J. Math. Anal. Appl. 324 (1), pp. 285–303.

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ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§18.36(i).

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17: Bibliography J

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Fonctions de Mathieu et fonctions propres de l’oscillateur relativiste. Ann. Fac. Sci. Toulouse Math. (6) 7 (3), pp. 465–495 (French).

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Notes:

Translated title: Mathieu functions and eigenfunctions of the relativistic oscillator.

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ISSN 0240-2963, Pdf, MathReview (Jean-Michel Ghez), ZentralBlatt

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7th item.

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A note on sampling expansion for a transform with parabolic cylinder kernel. Inform. Sci. 26 (2), pp. 155–158.

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ISSN 0020-0255, Document, MathReview, ZentralBlatt

Referenced by:

§12.16.

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Uniform asymptotic expansions for Meixner polynomials. Constr. Approx. 14 (1), pp. 113–150.

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ISSN 0176-4276, Document, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§18.24, §2.4(vi).

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Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions ${\overline{ps}}_n^r(\eta ,h)$ and ${\overline{qs}}_n^r(\eta ,h)$ for large $h$ . Proc. Roy. Soc. London Ser. A 321, pp. 545–555.

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FullText, MathReview, ZentralBlatt

Referenced by:

§30.9(ii).

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18: 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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§29.3(vii) Power Series

►For power-series expansions of the eigenvalues see Volkmer (2004b).