Liouville–Green approximation theorem

SearchAdvancedHelp

(0.001 seconds)

21—30 of 271 matching pages

21: 32.3 Graphics

… ►

►►

Figure 32.3.3: $w_k(x)$ for $-12\le x\le 0.73$ and $k=\mathrm{1.85185\hspace{0.33em}3}$, $\mathrm{1.85185\hspace{0.33em}5}$. The two graphs are indistinguishable when $x$ exceeds $-5.2$, approximately. … Magnify

… ►

►►

Figure 32.3.7: $u_k(x;-\frac{1}{2})$ for $-12\le x\le 4$ with $k=\mathrm{0.33554\hspace{0.33em}691}$, $\mathrm{0.33554\hspace{0.33em}692}$. …The parabolas $u^2+\frac{1}{2}x=0$, $u^2+\frac{1}{6}x=0$ are shown in black and green, respectively. Magnify

►

►►

Figure 32.3.8: $u_k(x;\frac{1}{2})$ for $-12\le x\le 4$ with $k=0.47442$, $0.47443$. …The curves $u^2+\frac{1}{3}x\pm \frac{1}{6}\sqrt{x^2+12}=0$ are shown in green and black, respectively. Magnify

►

►►

Figure 32.3.9: $u_k(x;\frac{3}{2})$ for $-12\le x\le 4$ with $k=0.38736$, $0.38737$. …The curves $u^2+\frac{1}{3}x\pm \frac{1}{6}\sqrt{x^2+24}=0$ are shown in green and black, respectively. Magnify

►

►►

Figure 32.3.10: $u_k(x;\frac{5}{2})$ for $-12\le x\le 4$ with $k=\mathrm{0.24499\hspace{0.33em}2}$, $\mathrm{0.24499\hspace{0.33em}3}$. …The curves $u^2+\frac{1}{3}x\pm \frac{1}{6}\sqrt{x^2+36}=0$ are shown in green and black, respectively. Magnify

22: 18.38 Mathematical Applications

… ►

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

►

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

ⓘ

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

…

23: 27.6 Divisor Sums

… ►

27.6.1

ⓘ

Symbols:

$\lambda \left(n\right)$: Liouville’s function, $d$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

…

24: 27.7 Lambert Series as Generating Functions

… ►

27.7.6 $$\sum _{n=1}^{\mathrm{\infty }}\lambda \left(n\right)\frac{x^n}{1-x^n}=\sum _{n=1}^{\mathrm{\infty }}{x}^{n^2}.$$

ⓘ

Symbols:

$\lambda \left(n\right)$: Liouville’s function, $n$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.7 and Ch.27

25: Bibliography P

… ►

Chebyshev series approximations for the zeros of the Bessel functions. J. Comput. Phys. 53 (1), pp. 188–192.

ⓘ

External Links:

ISSN 0021-9991, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.76(ii).

Encodings:

BibTeX

… ►

Approximation for the turning points of Bessel functions. J. Comput. Phys. 64 (1), pp. 253–257.

ⓘ

External Links:

ISSN 0021-9991, Document, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§10.76(ii).

Encodings:

BibTeX

… ►

On the maximum errors of polynomial approximations defined by interpolation and by least squares criteria. Comput. J. 9 (4), pp. 404–407.

ⓘ

External Links:

Document, MathReview, ZentralBlatt

Referenced by:

§3.11(ii).

Encodings:

BibTeX

… ►

Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 1 (4), pp. 372–379.

ⓘ

External Links:

ISSN 0098-3500, Document, GAMS (module 498; package TOMS), ZentralBlatt

Referenced by:

1st item, 4th item.

Encodings:

BibTeX

… ►

Numerical Solution of Sturm-Liouville Problems. Monographs on Numerical Analysis, The Clarendon Press, Oxford University Press, New York.

ⓘ

External Links:

ISBN 0-19-853415-9, MathReview (L. Greenberg), ZentralBlatt

Referenced by:

§3.7(iv).

Encodings:

BibTeX

…

26: 1.6 Vectors and Vector-Valued Functions

… ►

Green’s Theorem

… ►

Stokes’s Theorem

… ►

Gauss’s (or Divergence) Theorem

… ►

Green’s Theorem (for Volume)

…

27: 30.2 Differential Equations

… ►The Liouville normal form of equation (30.2.1) is …

28: 1.9 Calculus of a Complex Variable

… ►

DeMoivre’s Theorem

… ►

Jordan Curve Theorem

… ►

Cauchy’s Theorem

… ►

Liouville’s Theorem

… ►

Dominated Convergence Theorem

…

29: 15.16 Products

…

30: Bibliography J

… ►

REMES2 — a Fortran program to calculate rational minimax approximations to a given function. Technical Report Technical Report AECL-4210, Atomic Energy of Canada Limited. Chalk River Nuclear Laboratories, Chalk River, Ontario.

ⓘ

Referenced by:

§3.11(iii).

Encodings:

BibTeX

… ►

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.

ⓘ

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§30.9(ii).

Encodings:

BibTeX

… ►

On the simple cubic lattice Green function. Philos. Trans. Roy. Soc. London Ser. A 273, pp. 583–610.

ⓘ

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§31.17(ii).

Encodings:

BibTeX

►

On the cubic lattice Green functions. Proc. Roy. Soc. London Ser. A 445, pp. 463–477.

ⓘ

External Links:

ISSN 0962-8444, FullText, MathReview (Peter J. Forrester), ZentralBlatt

Referenced by:

§31.17(ii), §31.7(i).

Encodings:

BibTeX

…