Legendre equation

SearchAdvancedHelp

(0.005 seconds)

21—30 of 95 matching pages

21: 18.5 Explicit Representations

§18.5 Explicit Representations

… ►In this equation $w(x)$ is as in Table 18.3.1, (reproduced in Table 18.5.1), and $F(x)$, ${\kappa }_n$ are as in Table 18.5.1. … ►For corresponding formulas for Chebyshev, Legendre, and the Hermite ${\mathrm{𝐻𝑒}}_n$ polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). … ►

Legendre

… ►

22: 19.7 Connection Formulas

§19.7 Connection Formulas

… ►

Legendre’s Relation

… ►

Reciprocal-Modulus Transformation

… ►

Imaginary-Modulus Transformation

… ►

Imaginary-Argument Transformation

…

23: 14.17 Integrals

… ►

14.17.6 $${\int }_{-1}^1{𝖯}_l^m\left(x\right){𝖯}_n^m\left(x\right)dx=\frac{(n+m)!}{(n-m)!\left(n+\frac{1}{2}\right)}{\delta }_{l,n},$$

ⓘ

Symbols:

: Ferrers function of the first kind, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $x$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $l$: nonnegative integer

A&S Ref:

8.14.11 8.14.13

Referenced by:

§14.17(iii), Erratum (V1.0.17) for Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9)

Permalink:

http://dlmf.nist.gov/14.17.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

The Kronecker delta symbol has been moved furthest to the right.

See also:

Annotations for §14.17(iii), §14.17 and Ch.14

… ►

14.17.9 $${\int }_{-1}^1\frac{{𝖯}_n^l\left(x\right){𝖯}_n^{-m}\left(x\right)}{1-x^2}dx=\frac{{(-1)}^l}{l}{\delta }_{l,m},$$ $l>0$.

ⓘ

Symbols:

: Ferrers function of the first kind, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $x$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $l$: nonnegative integer

Referenced by:

§14.17(iii), Erratum (V1.0.17) for Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9)

Permalink:

http://dlmf.nist.gov/14.17.E9

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

The Kronecker delta symbol has been moved furthest to the right and ${(-1)}^l$ has been moved to the numerator of the fraction.

See also:

Annotations for §14.17(iii), §14.17 and Ch.14

…

24: T. Mark Dunster

… ►He has received a number of National Science Foundation grants, and has published numerous papers in the areas of uniform asymptotic solutions of differential equations, convergent WKB methods, special functions, quantum mechanics, and scattering theory. … ►

14 Legendre and Related Functions

…

25: 14.19 Toroidal (or Ring) Functions

… ►

14.19.2 $$P_{\nu -\frac{1}{2}}^{\mu }\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(\frac{1}{2}-\mu \right)}{{\pi }^{1/2}{\left(1-{\mathrm{e}}^{-2\xi }\right)}^{\mu }{\mathrm{e}}^{(\nu +(1/2))\xi }}𝐅(\frac{1}{2}-\mu ,\frac{1}{2}+\nu -\mu ;1-2\mu ;1-{\mathrm{e}}^{-2\xi }),$$ $\mu \ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $\mu$: general order, $\nu$: general degree and $\xi >0$: variable

Referenced by:

§14.19(ii), Erratum (V1.0.1) for Equation (14.19.2)

Permalink:

http://dlmf.nist.gov/14.19.E2

Encodings:

pMML, png, TeX

Errata (effective with 1.0.1):

Originally the argument to $𝐅$ was incorrect and the condition on $\mu$ too weak:

$$P_{\nu -\frac{1}{2}}^{\mu }\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(1-2\mu \right)2^{2\mu }}{\mathrm{\Gamma }\left(1-\mu \right){\left(1-{\mathrm{e}}^{-2\xi }\right)}^{\mu }{\mathrm{e}}^{(\nu +(1/2))\xi }}𝐅(\frac{1}{2}-\mu ,\frac{1}{2}+\nu -\mu ;1-2\mu ;{\mathrm{e}}^{-2\xi }),$$ $\mu \ne \frac{1}{2}$.

Also, the factor multiplying $𝐅$ was rewritten to clarify the poles, which determine the correct condition on $\mu$.

Reported 2010-11-02 by Alvaro Valenzuela

See also:

Annotations for §14.19(ii), §14.19 and Ch.14

…

26: 18.18 Sums

… ►

Legendre

… ►Equation (18.18.1) becomes … ►

Legendre

… ►and a similar pair of equations by symmetry; compare the second row in Table 18.6.1. … ►

Legendre and Chebyshev

…

27: Bibliography D

… ►

Convergent expansions for solutions of linear ordinary differential equations having a simple pole, with an application to associated Legendre functions. Stud. Appl. Math. 113 (3), pp. 245–270.

ⓘ

External Links:

ISSN 0022-2526, Document, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§14.15(iii), §2.8(i).

Encodings:

BibTeX

…

28: 10.19 Asymptotic Expansions for Large Order

… ►

10.19.9 $$\begin{array}{c}{H}_{\nu }^{(1)}\left(\nu +a{\nu }^{\frac{1}{3}}\right)\\ H_{\nu }^{(2)}\left(\nu +a{\nu }^{\frac{1}{3}}\right)\end{array}\}\sim \frac{2^{\frac{4}{3}}}{{\nu }^{\frac{1}{3}}}{\mathrm{e}}^{\mp \pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{P_k(a)}{{\nu }^{2k/3}}+\frac{2^{\frac{5}{3}}}{\nu }{\mathrm{e}}^{\pm \pi \mathrm{i}/3}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{Q_k(a)}{{\nu }^{2k/3}},$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $\nu$: complex parameter, $P_k(a)$: polynomial coefficient and $Q_k(a)$: polynomial coefficient

Permalink:

http://dlmf.nist.gov/10.19.E9

Encodings:

pMML, png, TeX

Correction (effective with 1.0.27):

Originally the polynomials $P_k(a)$, $Q_k(a)$ were incorrectly described as Legendre polynomials and integer-degree, zero-order associated Legendre functions of the second kind respectively.

See also:

Annotations for §10.19(iii), §10.19 and Ch.10

…

29: 22.16 Related Functions

… ►

Relation to Elliptic Integrals

… ►In Equations (22.16.21)–(22.16.23), … ►In Equations (22.16.24)–(22.16.26), . … ►

Relation to the Elliptic Integral $E(\varphi ,k)$

… ►

Definition

…

30: 22.19 Physical Applications

… ►

22.19.2 $$\sin \left(\frac{1}{2}\theta (t)\right)=\sin \left(\frac{1}{2}\alpha \right)\mathrm{sn}(t+K,\sin \left(\frac{1}{2}\alpha \right)),$$

ⓘ

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: sine function, $\theta (t)$: angular displacement and $\alpha$: initial displacement

Referenced by:

§22.19(i), §22.19(i), Erratum (V1.0.8) for Equation (22.19.2)

Permalink:

http://dlmf.nist.gov/22.19.E2

Encodings:

pMML, png, TeX

Errata (effective with 1.0.8):

Originally the first argument to the function $\mathrm{sn}$ was given incorrectly as $t$. The correct argument is $t+K$.

Reported 2014-03-05 by Svante Janson

See also:

Annotations for §22.19(i), §22.19 and Ch.22

…