Legendre equation

SearchAdvancedHelp

(0.003 seconds)

11—20 of 95 matching pages

11: 29.3 Definitions and Basic Properties

… ►For each pair of values of $\nu$ and $k$ there are four infinite unbounded sets of real eigenvalues $h$ for which equation (29.2.1) has even or odd solutions with periods $2K$ or $4K$. … … ►satisfies the continued-fraction equation … ►The quantity $H=2a_{\nu }^{2m+1}\left(k^2\right)-\nu (\nu +1)k^2$ satisfies equation (29.3.10) with … ►The quantity $H=2b_{\nu }^{2m+1}\left(k^2\right)-\nu (\nu +1)k^2$ satisfies equation (29.3.10) with …

12: Errata

… ►

14.8.3 $${𝖰}_{\nu }\left(x\right)=\frac{1}{2}\ln \left(\frac{2}{1-x}\right)-\gamma -\psi \left(\nu +1\right)+O\left(\left(1-x\right)\ln \left(1-x\right)\right),$$ $\nu \ne -1,-2,-3,\mathrm{\dots }$

The symbol $O\left(1-x\right)$ has been corrected to be $O\left(\left(1-x\right)\ln \left(1-x\right)\right)$.

Reported by Mark Ashbaugh on 2022-02-08

… ►

14.6.6 $${𝖯}_{\nu }^{-m}\left(x\right)={\left(1-x^2\right)}^{-m/2}{\int }_x^1\mathrm{\dots }{\int }_x^1{𝖯}_{\nu }\left(x\right){\left(dx\right)}^m$$

The right-hand side has been corrected by replacing the Legendre function $P_{\nu }\left(x\right)$ with the Ferrers function ${𝖯}_{\nu }\left(x\right)$.

… ►

Previously the exponents of the associated Legendre differential equation (14.2.2) at infinity were given incorrectly by $\{-\nu -1,\nu \}$. These were replaced by $\{\nu +1,-\nu \}$.

Reported by Hans Volkmer on 2019-01-30

… ►

14.5.14 $${𝖰}_{\nu }^{-1/2}\left(\cos \theta \right)={\left(\frac{\pi }{2\sin \theta }\right)}^{1/2}\frac{\cos \left(\left(\nu +\frac{1}{2}\right)\theta \right)}{\nu +\frac{1}{2}}$$

Originally this equation was incorrect because of a minus sign in front of the right-hand side.

Reported 2017-04-10 by André Greiner-Petter.

… ►

10.19.11 $$Q_3\left(a\right)=\frac{549}{28000}{a}^8-\frac{\mathrm{1\hspace{0.33em}10767}}{\mathrm{6\hspace{0.33em}93000}}{a}^5+\frac{79}{12375}{a}^2$$

Originally the first term on the right-hand side of this equation was written incorrectly as $-\frac{549}{28000}{a}^8$.

Reported 2015-03-16 by Svante Janson.

…

13: 14.6 Integer Order

… ►

14.6.6 $${𝖯}_{\nu }^{-m}\left(x\right)={\left(1-x^2\right)}^{-m/2}{\int }_x^1\mathrm{\dots }{\int }_x^1{𝖯}_{\nu }\left(x\right){\left(dx\right)}^m.$$

ⓘ

Symbols:

: Ferrers function of the first kind, $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

Referenced by:

§14.12(i), Erratum (V1.0.28) for Equation (14.6.6)

Permalink:

http://dlmf.nist.gov/14.6.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.28):

The right-hand side has been corrected by replacing the Legendre function $P_{\nu }\left(x\right)$ with the Ferrers function ${𝖯}_{\nu }\left(x\right)$.

See also:

Annotations for §14.6(ii), §14.6 and Ch.14

…

14: 29.8 Integral Equations

… ►

29.8.2 $$\mu w(z_1)w(z_2)w(z_3)={\int }_{-2K}^{2K}{𝖯}_{\nu }\left(x\right)w(z)dz,$$

ⓘ

Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $x$: real variable, $z$: complex variable, $k$: real parameter, $\nu$: real parameter, $w(z)$: solution and $\mu$

Referenced by:

§29.8

Permalink:

http://dlmf.nist.gov/29.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

…

15: 14.5 Special Values

… ►

14.5.3 $${𝖰}_{\nu }^{\mu }\left(0\right)=-\frac{2^{\mu -1}{\pi }^{1/2}\sin \left(\frac{1}{2}(\nu +\mu )\pi \right)\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{1}{2}\right)}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +1\right)},$$ $\nu +\mu \ne -1,-2,-3,\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.6.2

Referenced by:

Erratum (V1.1.3) for Equations (14.5.3), (14.5.4), Erratum (V1.1.3) for Equations (14.5.3), (14.5.4)

Permalink:

http://dlmf.nist.gov/14.5.E3

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The constraint has been corrected to exclude all negative integers.

See also:

Annotations for §14.5(i), §14.5 and Ch.14

►

14.5.4 $${\frac{d{𝖰}_{\nu }^{\mu }\left(x\right)}{dx}|}_{x=0}=\frac{2^{\mu }{\pi }^{1/2}\cos \left(\frac{1}{2}(\nu +\mu )\pi \right)\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +1\right)}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +\frac{1}{2}\right)},$$ $\nu +\mu \ne -1,-2,-3,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.6.4

Referenced by:

Erratum (V1.1.3) for Equations (14.5.3), (14.5.4), Erratum (V1.1.3) for Equations (14.5.3), (14.5.4)

Permalink:

http://dlmf.nist.gov/14.5.E4

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The constraint has been corrected to exclude all negative integers.

See also:

Annotations for §14.5(i), §14.5 and Ch.14

… ►

14.5.14 $${𝖰}_{\nu }^{-1/2}\left(\cos \theta \right)={\left(\frac{\pi }{2\sin \theta }\right)}^{1/2}\frac{\cos \left(\left(\nu +\frac{1}{2}\right)\theta \right)}{\nu +\frac{1}{2}}.$$

ⓘ

Symbols:

: Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\nu$: general degree and : variable

A&S Ref:

8.6.15 (corrected)

Referenced by:

Erratum (V1.0.16) for Equation (14.5.14)

Permalink:

http://dlmf.nist.gov/14.5.E14

Encodings:

pMML, png, TeX

Errata (effective with 1.0.16):

Originally this equation was incorrect because of a minus sign in front of the right-hand side.

Reported 2017-04-10 by André Greiner-Petter

See also:

Annotations for §14.5(iii), §14.5 and Ch.14

…

16: 19.2 Definitions

… ►

19.2.8_1 $$K^{\prime }\left(k\right)={\int }_0^1\frac{dt}{\sqrt{1-t^2}\sqrt{1-(1-k^2)t^2}},$$

ⓘ

Defines:

$K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind

Symbols:

$dx$: differential of $x$ , $\int$: integral and $k$: real or complex modulus

Referenced by:

§19.2(ii)

Permalink:

http://dlmf.nist.gov/19.2.E8_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.10):

This equation was added.

See also:

Annotations for §19.2(ii), §19.2 and Ch.19

►

19.2.8_2 $$E^{\prime }\left(k\right)={\int }_0^1\frac{\sqrt{1-(1-k^2)t^2}}{\sqrt{1-t^2}}dt,$$

ⓘ

Defines:

$E^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the second kind

Symbols:

$dx$: differential of $x$ , $\int$: integral and $k$: real or complex modulus

Referenced by:

§19.2(ii)

Permalink:

http://dlmf.nist.gov/19.2.E8_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.10):

This equation was added.

See also:

Annotations for §19.2(ii), §19.2 and Ch.19

…

17: Frank W. J. Olver

… ►He is particularly known for his extensive work in the study of the asymptotic solution of differential equations, i. …, the behavior of solutions as the independent variable, or some parameter, tends to infinity, and in the study of the particular solutions of differential equations known as special functions (e. …, Bessel functions, hypergeometric functions, Legendre functions). …

18: 19.5 Maclaurin and Related Expansions

… ►

19.5.4_1 $$F(\varphi ,k)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_m{\sin }^{2m+1}\varphi }{(2m+1)m!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{m+\frac{1}{2},\frac{1}{2}}{m+\frac{3}{2}};{\sin }^2\varphi )k^{2m}=\sin \varphi F_1(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};{\sin }^2\varphi ,k^2{\sin }^2\varphi ),$$

ⓘ

Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $\varphi$: real or complex argument and $k$: real or complex modulus

Source:

Carlson (1961b, (2.8))

Referenced by:

§19.5, Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/19.5.E4_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §19.5 and Ch.19

►

19.5.4_2 $$E(\varphi ,k)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(-\frac{1}{2}\right)}_m{\sin }^{2m+1}\varphi }{(2m+1)m!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{m+\frac{1}{2},\frac{1}{2}}{m+\frac{3}{2}};{\sin }^2\varphi )k^{2m}=\sin \varphi F_1(\frac{1}{2};\frac{1}{2},-\frac{1}{2};\frac{3}{2};{\sin }^2\varphi ,k^2{\sin }^2\varphi ),$$

ⓘ

Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $\varphi$: real or complex argument and $k$: real or complex modulus

Source:

Carlson (1961b, (2.9))

Referenced by:

§19.5, Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/19.5.E4_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §19.5 and Ch.19

►

19.5.4_3 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_m{\sin }^{2m+1}\varphi }{(2m+1)m!}{F}_1(m+\frac{1}{2};\frac{1}{2},1;m+\frac{3}{2};{\sin }^2\varphi ,{\alpha }^2{\sin }^2\varphi )k^{2m},$$

ⓘ

Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $\varphi$: real or complex argument, $k$: real or complex modulus and ${\alpha }^2$: real or complex parameter

Source:

Carlson (1961b, (2.11))

Referenced by:

§19.5, Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/19.5.E4_3

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §19.5 and Ch.19

…

19: 14.13 Trigonometric Expansions

… ►

14.13.1 $${𝖯}_{\nu }^{\mu }\left(\cos \theta \right)=\frac{2^{\mu +1}{(\sin \theta )}^{\mu }}{{\pi }^{1/2}}\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\nu +\mu +k+1\right)}{\mathrm{\Gamma }\left(\nu +k+\frac{3}{2}\right)}\frac{{\left(\mu +\frac{1}{2}\right)}_k}{k!}\sin \left((\nu +\mu +2k+1)\theta \right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, $\mu$: general order, $\nu$: general degree and : variable

Source:

Volkmer (2021)

A&S Ref:

8.7.1

Referenced by:

§14.13, Erratum (V1.1.3) for Equations (14.13.1), (14.13.2)

Permalink:

http://dlmf.nist.gov/14.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.13 and Ch.14

►

14.13.2 $${𝖰}_{\nu }^{\mu }\left(\cos \theta \right)={\pi }^{1/2}{2}^{\mu }{(\sin \theta )}^{\mu }\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\nu +\mu +k+1\right)}{\mathrm{\Gamma }\left(\nu +k+\frac{3}{2}\right)}\frac{{\left(\mu +\frac{1}{2}\right)}_k}{k!}\cos \left((\nu +\mu +2k+1)\theta \right).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, $\mu$: general order, $\nu$: general degree and : variable

Source:

Cohl et al. (2021, Theorem 6.2)

A&S Ref:

8.7.2

Referenced by:

§14.13, Erratum (V1.1.3) for Equations (14.13.1), (14.13.2)

Permalink:

http://dlmf.nist.gov/14.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.13 and Ch.14

…

20: Bibliography S

… ►

Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory. National Bureau of Standards Applied Mathematics Series, No. 19, U. S. Government Printing Office, Washington, D.C..

ⓘ

External Links:

MathReview, ZentralBlatt

Referenced by:

§13.23(iv), Ch.14, §31.3(iii).

Encodings:

BibTeX

…