expansions in series of eigenfunctions

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1: 28.30 Expansions in Series of Eigenfunctions

§28.30 Expansions in Series of Eigenfunctions

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2: 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. …

3: 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. … ►It will be observed that the present formulation of the Taylor-series method permits considerable parallelism in the computation, both for initial-value and boundary-value problems. … ►General methods for boundary-value problems for ordinary differential equations are given in Ascher et al. (1995). … ►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. …

4: 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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External Links:

FullText, MathReview

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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5: Bibliography M

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Siegel’s modular forms and Dirichlet series. Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin.

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External Links:

Document, MathReview (K.-B. Gundlach), ZentralBlatt

Referenced by:

§35.4(ii).

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On the convergence of the Chebyshev series for functions possessing a singularity in the range of representation. SIAM J. Numer. Anal. 3 (3), pp. 390–409.

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ISSN 1095-7170, Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§3.11(ii).

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A $q$-analog of hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ and invariant $G$-functions. Adv. in Math. 58 (1), pp. 1–60.

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ISSN 0001-8708, Document, MathReview (Robert A. Gustafson), ZentralBlatt

Referenced by:

§17.15.

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A $q$-analog of the Gauss summation theorem for hypergeometric series in $U(n)$ . Adv. in Math. 72 (1), pp. 59–131.

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ISSN 0001-8708, Document, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.15.

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A $q$-analog of a Whipple’s transformation for hypergeometric series in $U(n)$ . Adv. Math. 108 (1), pp. 1–76.

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ISSN 0001-8708, Document, MathReview, ZentralBlatt

Referenced by:

§17.15.

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6: Bibliography R

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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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Elliptic hypergeometric series on root systems. Adv. Math. 181 (2), pp. 417–447.

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ISSN 0001-8708, Document, MathReview, ZentralBlatt

Referenced by:

§20.11(iii).

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Sources in the development of mathematics. Cambridge University Press, Cambridge.

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Notes:

Infinite Series and Products from the Fifteenth to the Twenty-first Century

External Links:

ISBN 978-0-521-11470-7, Document, Link, MathReview (Mehdi Hassani)

Referenced by:

Profile Ranjan Roy.

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Functional Analysis. McGraw-Hill Book Co., New York.

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Notes:

McGraw-Hill Series in Higher Mathematics

External Links:

MathReview (F. Smithies), ZentralBlatt

Referenced by:

§2.6(i).

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Principles of Mathematical Analysis. 3rd edition, McGraw-Hill Book Co., New York.

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Notes:

International Series in Pure and Applied Mathematics

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.4(iii).

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7: 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 ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$, $n=m,m+1,m+2,\mathrm{\dots }$. … ► ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ has exactly $n-m$ zeros in the interval . ►

§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 $-1\le x\le 1$. …

8: 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.

External Links:

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

Referenced by:

7th item.

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Bounds on Dawson’s integral occurring in the analysis of a line distribution network for electric vehicles. Eurandom Preprint Series Technical Report 14, Eurandom, Eindhoven, The Netherlands.

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Notes:

ISSN 1389-2355

External Links:

FullText

Referenced by:

§7.8, §7.8.

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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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Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, FL.

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Notes:

A second edition was published in 2010.

External Links:

ISBN 1-58488-296-4, MathReview (Jian Hong Wu), ZentralBlatt

Referenced by:

D. S. Jones, M. J. Plank, and B. D. Sleeman (2010).

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Calculus of Finite Differences. Hungarian Agent Eggenberger Book-Shop, Budapest.

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Third edition published in 1965 by Chelsea Publishing Co., New York, and American Mathematical Society, AMS Chelsea Book Series, Providence, RI

External Links:

MathReview (W. E. Milne), ZentralBlatt

Referenced by:

§26.1, §26.1, §26.8(vii), §5.16, Notations S.

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9: Bibliography K

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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)

Referenced by:

§17.2(i).

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Electromagnetic Fields and Relativistic Particles. International Series in Pure and Applied Physics, McGraw-Hill Book Co., New York.

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External Links:

ISBN 0-07-035264-X, MathReview (George Zipfel, Jr.)

Referenced by:

§1.17(ii), §10.73(ii).

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HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively $q$-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

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Notes:

Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.

External Links:

Code, Document, MathReview (G. Gasper), ZentralBlatt

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1st item.

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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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External Links:

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

Referenced by:

§18.36(i).

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10: 29.3 Definitions and Basic Properties

… ►The eigenfunctions corresponding to the eigenvalues of §29.3(i) are denoted by ${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)$, ${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)$. …In this table the nonnegative integer $m$ corresponds to the number of zeros of each Lamé function in $(0,K)$, whereas the superscripts $2m$, $2m+1$, or $2m+2$ correspond to the number of zeros in $[0,2K)$. ►

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).