# expansions in series of eigenfunctions

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##### 1: 28.30 Expansions in Series of Eigenfunctions

###### §28.30 Expansions in Series of Eigenfunctions

…##### 2: Howard S. Cohl

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βΊCohl 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.
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##### 3: 18.3 Definitions

###### §18.3 Definitions

… βΊAs eigenfunctions of second order differential operators
(*Bochner’s theorem*, Bochner (1929)).
See the differential equations
$A\beta \x81\u2019(x)\beta \x81\u2019{p}_{n}^{\prime \prime}\beta \x81\u2018(x)+B\beta \x81\u2019(x)\beta \x81\u2019{p}_{n}^{\prime}\beta \x81\u2018(x)+{\mathrm{\Xi \xbb}}_{n}\beta \x81\u2019{p}_{n}\beta \x81\u2018(x)=0$,
in Table 18.8.1.

##### 4: 3.7 Ordinary Differential Equations

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βΊFor classification of singularities of (3.7.1) and expansions of solutions in the neighborhoods of singularities, see §2.7.
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βΊIt 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.
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βΊGeneral methods for boundary-value problems for ordinary differential equations are given in Ascher et al. (1995).
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βΊThe eigenvalues ${\mathrm{\Xi \xbb}}_{k}$ are simple, that is, there is only one corresponding eigenfunction (apart from a normalization factor), and when ordered increasingly the eigenvalues satisfy
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βΊThe 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.
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##### 5: Bibliography V

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βΊ
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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βΊ
Expansion of vacuum magnetic fields in toroidal harmonics.
Comput. Phys. Comm. 81 (1-2), pp. 74–90.
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βΊ
Symbolic evaluation of coefficients in Airy-type asymptotic expansions.
J. Math. Anal. Appl. 269 (1), pp. 317–331.
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βΊ
Expansions in products of Heine-Stieltjes polynomials.
Constr. Approx. 15 (4), pp. 467–480.
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βΊ
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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##### 6: Bibliography M

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βΊ
Siegel’s modular forms and Dirichlet series.
Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin.
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βΊ
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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βΊ
A $q$-analog of hypergeometric series well-poised in
$\mathrm{\pi \x9d\x91\x86\pi \x9d\x91\x88}\beta \x81\u2019(n)$ and invariant $G$-functions.
Adv. in Math. 58 (1), pp. 1–60.
βΊ
A $q$-analog of the Gauss summation theorem for hypergeometric series in
$U\beta \x81\u2019(n)$
.
Adv. in Math. 72 (1), pp. 59–131.
βΊ
A $q$-analog of a Whipple’s transformation for hypergeometric series in
$U\beta \x81\u2019(n)$
.
Adv. Math. 108 (1), pp. 1–76.
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##### 7: Bibliography T

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βΊ
Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions.
SIAM J. Math. Anal. 21 (1), pp. 241–261.
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Eigenfunction Expansions Associated with Second-Order Differential Equations.
Clarendon Press, Oxford.
βΊ
Eigenfunction Expansions Associated with Second Order Differential Equations, Part 2, Partial Differential Equations.
Clarendon Press, Oxford.
βΊ
Eigenfunction expansions associated with second-order differential equations. Part I.
Second edition, Clarendon Press, Oxford.
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βΊ
Iterative Methods for the Solution of Equations.
Prentice-Hall Series in Automatic Computation, Prentice-Hall Inc., Englewood Cliffs, N.J..
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##### 8: 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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Elliptic hypergeometric series on root systems.
Adv. Math. 181 (2), pp. 417–447.
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βΊ
Sources in the development of mathematics.
Cambridge University Press, Cambridge.
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βΊ
Functional Analysis.
McGraw-Hill Book Co., New York.
βΊ
Principles of Mathematical Analysis.
3rd edition, McGraw-Hill Book Co., New York.
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##### 9: 30.4 Functions of the First Kind

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βΊThe eigenfunctions of (30.2.1) that correspond to the eigenvalues ${\mathrm{\Xi \xbb}}_{n}^{m}\beta \x81\u2018\left({\mathrm{\Xi \xb3}}^{2}\right)$ are denoted by ${\mathrm{\pi \x9d\x96\u2015\pi \x9d\x97\x8c}}_{n}^{m}\beta \x81\u2018(x,{\mathrm{\Xi \xb3}}^{2})$, $n=m,m+1,m+2,\mathrm{\dots}$.
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βΊ
${\mathrm{\pi \x9d\x96\u2015\pi \x9d\x97\x8c}}_{n}^{m}\beta \x81\u2018(x,{\mathrm{\Xi \xb3}}^{2})$ has exactly $n-m$ zeros in the interval $$.
βΊ

###### §30.4(iii) Power-Series Expansion

… βΊThe expansion (30.4.7) converges in the norm of ${L}^{2}\beta \x81\u2018(-1,1)$, 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 $-1\le x\le 1$. …##### 10: Bibliography J

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βΊ
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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βΊ
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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βΊ
A note on sampling expansion for a transform with parabolic cylinder kernel.
Inform. Sci. 26 (2), pp. 155–158.
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βΊ
Differential equations and mathematical biology.
Chapman & Hall/CRC Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, FL.
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βΊ
Calculus of Finite Differences.
Hungarian Agent Eggenberger Book-Shop, Budapest.
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