Liouville%E2%80%93Green approximation theorem

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21: Bibliography G

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An extended class of orthogonal polynomials defined by a Sturm-Liouville problem. J. Math. Anal. Appl. 359 (1), pp. 352–367.

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External Links:

ISSN 0022-247X, Document, Link, MathReview (Li Zhong Peng)

Referenced by:

§18.36(vi).

Encodings:

BibTeX

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Superstring Theory: Introduction, Vol. 1. 2nd edition, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-35752-7, MathReview, ZentralBlatt

Referenced by:

§20.12(ii), 3rd item.

Encodings:

BibTeX

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Superstring Theory: Loop Amplitudes, Anomalies and Phenomenolgy, Vol. 2. 2nd edition, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-35753-5, MathReview, ZentralBlatt

Referenced by:

§20.12(ii).

Encodings:

BibTeX

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General form of the quantum-defect theory. Phys. Rev. A 19 (4), pp. 1485–1509.

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External Links:

Document

Referenced by:

item Greene et al. (1979):, Notations F, Notations F, Notations G.

Encodings:

BibTeX

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Mathematics for the Analysis of Algorithms. Progress in Computer Science, Vol. 1, Birkhäuser Boston, Boston, MA.

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External Links:

MathReview (Ernst-Erich Doberkat), ZentralBlatt

Referenced by:

§5.17.

Encodings:

BibTeX

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22: 2.8 Differential Equations with a Parameter

… ►In Case III $f(z)$ has a simple pole at $z_0$ and ${(z-z_0)}^{2}g(z)$ is analytic at $z_0$. … ►First we apply the Liouville transformation (§1.13(iv)) to (2.8.1). … ►In Case III the approximating equation is … ►For connection formulas for Liouville–Green approximations across these transition points see Olver (1977b, a, 1978). … ►For examples of uniform asymptotic approximations in terms of Whittaker functions with fixed second parameter see §18.15(i) and §28.8(iv). …

23: 3.8 Nonlinear Equations

… ►For real functions $f(x)$ the sequence of approximations to a real zero $\xi$ will always converge (and converge quadratically) if either: … ►Inverse linear interpolation (§3.3(v)) is used to obtain the first approximation: … ►Initial approximations to the zeros can often be found from asymptotic or other approximations to $f(z)$, or by application of the phase principle or Rouché’s theorem; see §1.10(iv). … … ►For describing the distribution of complex zeros of solutions of linear homogeneous second-order differential equations by methods based on the Liouville–Green (WKB) approximation, see Segura (2013). …

24: 15.16 Products

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25: 1.9 Calculus of a Complex Variable

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DeMoivre’s Theorem

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Jordan Curve Theorem

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Cauchy’s Theorem

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Liouville’s Theorem

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Dominated Convergence Theorem

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26: 3.7 Ordinary Differential Equations

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§3.7(iv) Sturm–Liouville Eigenvalue Problems

… ►The Sturm–Liouville eigenvalue problem is the construction of a nontrivial solution of the system …

27: 18.38 Mathematical Applications

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§18.38(i) Classical OP’s: Numerical Analysis

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Approximation Theory

… ►For these results and applications in approximation theory see §3.11(ii) and Mason and Handscomb (2003, Chapter 3), Cheney (1982, p. 108), and Rivlin (1969, p. 31). … ►The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. … ►

18.38.3 $$\sum _{m=0}^n{P}_m^{(\alpha ,0)}\left(x\right)=\frac{{\left(\alpha +2\right)}_n}{n!}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-n,n+\alpha +2,\frac{1}{2}(\alpha +1)}{\alpha +1,\frac{1}{2}(\alpha +3)};\frac{1}{2}(1-x))\ge 0,$$ $x\ge -1$, $\alpha \ge -2$, $n=0,1,\mathrm{\dots }$,

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Askey and Gasper (1976, Theorem 3, p.717)(proved)

Referenced by:

§18.14(iv), §18.18(viii), Erratum (V1.2.0) for Equation (18.38.3)

Permalink:

http://dlmf.nist.gov/18.38.E3

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

This equation was updated to include the value of the sum in terms of the ${}_3{}^{}F_2^{}$ function. Also the constraint was previously $\alpha >-1$.

See also:

Annotations for §18.38(ii), §18.38(ii), §18.38 and Ch.18

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28: 1.15 Summability Methods

… ►For $\mathrm{\Re }\alpha >0$ and $x\ge 0$, the Riemann-Liouville fractional integral of order $\alpha$ is defined by … ►

§1.15(viii) Tauberian Theorems

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29: 2.6 Distributional Methods

§2.6 Distributional Methods

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§2.6(ii) Stieltjes Transform

… ►Corresponding results for the generalized Stieltjes transform … ►The Riemann–Liouville fractional integral of order $\mu$ is defined by … ►For rigorous derivations of these results and also order estimates for ${\delta }_n(x)$, see Wong (1979) and Wong (1989, Chapter 6).

30: 27.3 Multiplicative Properties

… ►Examples are $\lfloor 1/n\rfloor$ and $\lambda \left(n\right)$, and the Dirichlet characters, defined in §27.8. …