expansions of solutions in series of

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11: 28.34 Methods of Computation

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Summation of the power series in §§28.6(i) and 28.15(i) when $\left|q\right|$ is small.

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§28.34(iii) Floquet Solutions

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Summation of the power series in §§28.6(ii) and 28.15(ii) when $\left|q\right|$ is small.

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Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

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Numerical summation of the expansions in series of Bessel functions (28.24.1)–(28.24.13). These series converge quite rapidly for a wide range of values of $q$ and $z$.

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12: 2.9 Difference Equations

… ►Often $f(n)$ and $g(n)$ can be expanded in series … ►As in the case of differential equations (§§2.7(iii), 2.7(iv)) recessive solutions are unique and dominant solutions are not; furthermore, one member of a numerically satisfactory pair has to be recessive. … ►For asymptotic expansions in inverse factorial series see Olde Daalhuis (2004a). … ►But there is an independent solution … ►For discussions of turning points, transition points, and uniform asymptotic expansions for solutions of linear difference equations of the second order see Wang and Wong (2003, 2005). …

13: 18.40 Methods of Computation

… ►Usually, however, other methods are more efficient, especially the numerical solution of difference equations (§3.6) and the application of uniform asymptotic expansions (when available) for OP’s of large degree. For applications in which the OP’s appear only as terms in series expansions (compare §18.18(i)) the need to compute them can be avoided altogether by use instead of Clenshaw’s algorithm (§3.11(ii)) and its straightforward generalization to OP’s other than Chebyshev. … … ►Given the power moments, ${\mu }_n={\int }_a^b{x}^{n}d\mu (x)$, $n=0,1,2,\mathrm{\dots }$, can these be used to find a unique $\mu (x)$, a non-decreasing, real, function of $x$, in the case that the moment problem is determined? Should a unique solution not exist the moment problem is then indeterminant. … ►in which …

14: 10.74 Methods of Computation

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§10.74(i) Series Expansions

►The power-series expansions given in §§10.2 and 10.8, together with the connection formulas of §10.4, can be used to compute the Bessel and Hankel functions when the argument $x$ or $z$ is sufficiently small in absolute value. … ►In other circumstances the power series are prone to slow convergence and heavy numerical cancellation. … ►Furthermore, the attainable accuracy can be increased substantially by use of the exponentially-improved expansions given in §10.17(v), even more so by application of the hyperasymptotic expansions to be found in the references in that subsection. … ►In the interval , $J_{\nu }\left(x\right)$ needs to be integrated in the forward direction and $Y_{\nu }\left(x\right)$ in the backward direction, with initial values for the former obtained from the power-series expansion (10.2.2) and for the latter from asymptotic expansions (§§10.17(i) and 10.20(i)). …

15: 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. … ►

§3.7(ii) Taylor-Series Method: Initial-Value Problems

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§3.7(iii) Taylor-Series Method: Boundary-Value Problems

… ►General methods for boundary-value problems for ordinary differential equations are given in Ascher et al. (1995). … ►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. …

16: 13.2 Definitions and Basic Properties

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Standard Solutions

►The first two standard solutions are: … ►The series (13.2.2) and (13.2.3) converge for all $z\in ℂ$. … ►

§13.2(v) Numerically Satisfactory Solutions

►Fundamental pairs of solutions of (13.2.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are …

17: 28.30 Expansions in Series of Eigenfunctions

§28.30 Expansions in Series of Eigenfunctions

… ►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 ►

28.30.1 $$w_m^{\prime \prime }+({\widehat{\lambda }}_m+Q(x))w_m=0,$$

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Symbols:

$m$: integer, $x$: real variable, $w(z)$: Mathieu’s equation solution and ${\widehat{\lambda }}_m$: characteristic values

Permalink:

http://dlmf.nist.gov/28.30.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.30(i), §28.30 and Ch.28

… ►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 ►

28.30.3 $$f(x)=\sum _{m=0}^{\mathrm{\infty }}{f}_m{w}_m(x),$$

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Symbols:

$m$: integer, $x$: real variable and $w(z)$: Mathieu’s equation solution

Permalink:

http://dlmf.nist.gov/28.30.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.30(i), §28.30 and Ch.28

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18: 18.38 Mathematical Applications

… ►In consequence, expansions of functions that are infinitely differentiable on $[-1,1]$ in series of Chebyshev polynomials usually converge extremely rapidly. … ►Linear ordinary differential equations can be solved directly in series of Chebyshev polynomials (or other OP’s) by a method originated by Clenshaw (1957). … ►

Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions

… ►has a solution … ►Hermite EOP’s appear in solutions of a rationally modified Schrödinger equation in §18.39. …

19: 2.7 Differential Equations

… ►All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients. … ►Hence unless the series (2.7.8) terminate (in which case the corresponding ${\mathrm{\Lambda }}_j$ is zero) they diverge. … ►Although the expansions (2.7.14) apply only in the sectors (2.7.15) and (2.7.16), each solution $w_j(z)$ can be continued analytically into any other sector. … ►Note that the coefficients in the expansions (2.7.12), (2.7.13) for the “late” coefficients, that is, $a_{s,1}$, $a_{s,2}$ with $s$ large, are the “early” coefficients $a_{j,2}$, $a_{j,1}$ with $j$ small. … ►In theory either pair may be used to construct any other solution …

20: Bibliography L

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New method to obtain small parameter power series expansions of Mathieu radial and angular functions. Math. Comp. 78 (265), pp. 255–274.

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External Links:

ISSN 0025-5718, Document, FullText, MathReview (Junesang Choi), ZentralBlatt

Referenced by:

§28.24, §28.24.

Encodings:

BibTeX

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Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics. J. Math. Phys. 27 (5), pp. 1238–1265.

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External Links:

ISSN 0022-2488, Document, MathReview (Basilis C. Xanthopoulos), ZentralBlatt

Referenced by:

§30.12, §31.17(ii).

Encodings:

BibTeX

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New series expansions for the confluent hypergeometric function $M(a,b,z)$ . Appl. Math. Comput. 235, pp. 26–31.

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External Links:

ISSN 0096-3003, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§13.11, §13.11.

Encodings:

BibTeX

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New series expansions of the Gauss hypergeometric function. Adv. Comput. Math. 39 (2), pp. 349–365.

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External Links:

ISSN 1019-7168, Document, FullText, MathReview (Jochen Denzler), ZentralBlatt

Referenced by:

§15.19(i), §15.19(i).

Encodings:

BibTeX

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Expansion of the confluent hypergeometric function in series of Bessel functions. Math. Tables Aids Comput. 13 (68), pp. 261–271.

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External Links:

FullText, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§13.24(ii).

Encodings:

BibTeX

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