Legendre equation

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1: 14.21 Definitions and Basic Properties

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§14.21(i) Associated Legendre Equation

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14.21.1 $$\left(1-z^2\right)\frac{d^{2}w}{{dz}^2}-2z\frac{dw}{dz}+\left(\nu (\nu +1)-\frac{{\mu }^2}{1-z^2}\right)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.1.1

Referenced by:

§14.21(ii), §14.29, 2nd item

Permalink:

http://dlmf.nist.gov/14.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.21(i), §14.21 and Ch.14

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§14.21(ii) Numerically Satisfactory Solutions

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2: 14.2 Differential Equations

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§14.2(i) Legendre’s Equation

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§14.2(ii) Associated Legendre Equation

… ►Ferrers functions and the associated Legendre functions are related to the Legendre functions by the equations ${𝖯}_{\nu }^0\left(x\right)={𝖯}_{\nu }\left(x\right)$, ${𝖰}_{\nu }^0\left(x\right)={𝖰}_{\nu }\left(x\right)$, $P_{\nu }^0\left(x\right)=P_{\nu }\left(x\right)$, $Q_{\nu }^0\left(x\right)=Q_{\nu }\left(x\right)$, ${𝑸}_{\nu }^0\left(x\right)={𝑸}_{\nu }\left(x\right)=Q_{\nu }\left(x\right)/\mathrm{\Gamma }\left(\nu +1\right)$. … ►

§14.2(iii) Numerically Satisfactory Solutions

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14.2.7 $$𝒲\left\{P_{\nu }^{-\mu }\left(x\right),P_{\nu }^{\mu }\left(x\right)\right\}=𝒲\left\{{𝖯}_{\nu }^{-\mu }\left(x\right),{𝖯}_{\nu }^{\mu }\left(x\right)\right\}=\frac{2\sin \left(\mu \pi \right)}{\pi \left(1-x^2\right)},$$

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Symbols:

: Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

(14.2.7), §14.21(iii), Erratum (V1.0.9) for Equation (14.2.7)

Permalink:

http://dlmf.nist.gov/14.2.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.0.9):

(14.2.7) has been expanded to provide the Wronskian for Ferrers functions as well as for associated Legendre functions.

Suggested 2014-05-22 by Howard Cohl

See also:

Annotations for §14.2(iv), §14.2 and Ch.14

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3: 14.29 Generalizations

… ►For inhomogeneous versions of the associated Legendre equation, and properties of their solutions, see Babister (1967, pp. 252–264).

4: 18.8 Differential Equations

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Table 18.8.1: Classical OP’s: differential equations $A(x)f^{\prime \prime }(x)+B(x)f^{\prime }(x)+C(x)f(x)+{\lambda }_{n}f(x)=0$.

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#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	${\lambda }_n$
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5: 14.31 Other Applications

… ►The conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^m\left(x\right)$ appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). … ►

§14.31(iii) Miscellaneous

►Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). …

6: 19.4 Derivatives and Differential Equations

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19.4.3 $$\frac{d^{2}E\left(k\right)}{{dk}^2}=-\frac{1}{k}\frac{dK\left(k\right)}{dk}=\frac{k_{}^{\prime }{}_{}{}^{2}K\left(k\right)-E\left(k\right)}{k^{2}k_{}^{\prime }{}_{}{}^2},$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/19.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(i), §19.4 and Ch.19

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19.4.6 $$\frac{\partial E(\varphi ,k)}{\partial k}=\frac{E(\varphi ,k)-F(\varphi ,k)}{k},$$

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.30(i)

Permalink:

http://dlmf.nist.gov/19.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(i), §19.4 and Ch.19

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§19.4(ii) Differential Equations

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19.4.8 $$(kk_{}^{\prime }{}_{}{}^2{D}_k^2+(1-3k^2)D_k-k)F(\varphi ,k)=\frac{-k\sin \varphi \cos \varphi }{{(1-k^2{\sin }^2\varphi )}^{3/2}},$$

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Symbols:

$\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $D_k$: differential operator

Referenced by:

§19.4(ii)

Permalink:

http://dlmf.nist.gov/19.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(ii), §19.4 and Ch.19

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19.4.9 $$(kk_{}^{\prime }{}_{}{}^2{D}_k^2+k_{}^{\prime }{}_{}{}^2{D}_k+k)E(\varphi ,k)=\frac{k\sin \varphi \cos \varphi }{\sqrt{1-k^2{\sin }^2\varphi }}.$$

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Symbols:

$\cos z$: cosine function, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $D_k$: differential operator

Permalink:

http://dlmf.nist.gov/19.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(ii), §19.4 and Ch.19

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7: 14.3 Definitions and Hypergeometric Representations

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14.3.6

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $𝐅(a,b;c;z)$ 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 : real variable

Referenced by:

§14.21(i), §14.21(iii), §14.3(ii), §14.3(iv), §15.9(iv)

Permalink:

http://dlmf.nist.gov/14.3.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(ii), §14.3(ii), §14.3 and Ch.14

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14.3.8

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $m$: nonnegative integer, $\nu$: general degree and : real variable

Referenced by:

§14.3(ii)

Permalink:

http://dlmf.nist.gov/14.3.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(ii), §14.3(ii), §14.3 and Ch.14

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14.3.9

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $𝐅(a,b;c;z)$ 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 : real variable

Permalink:

http://dlmf.nist.gov/14.3.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(ii), §14.3(ii), §14.3 and Ch.14

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14.3.11

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Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\mu$: general order, $\nu$: general degree, : real variable and $w_2$: function $w_1$,

A&S Ref:

8.1.4 (modified)

Referenced by:

§10.59, §14.23, §14.3(iii), §14.5(i)

Permalink:

http://dlmf.nist.gov/14.3.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(iii), §14.3 and Ch.14

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14.3.15

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $𝐅(a,b;c;z)$ 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 : real variable

Referenced by:

§14.21(iii)

Permalink:

http://dlmf.nist.gov/14.3.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(iii), §14.3 and Ch.14

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8: 30.2 Differential Equations

… ►If $\gamma =0$, Equation (30.2.1) is the associated Legendre differential equation; see (14.2.2). …

9: 14.5 Special Values

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14.5.5 $${𝖯}_0\left(x\right)=P_0\left(x\right)=1,$$

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Symbols:

${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind and $x$: real variable

A&S Ref:

8.4.1

Permalink:

http://dlmf.nist.gov/14.5.E5

Encodings:

pMML, png, TeX

See also:

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

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14.5.6 $${𝖯}_1\left(x\right)=P_1\left(x\right)=x.$$

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Symbols:

${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind and $x$: real variable

A&S Ref:

8.4.3

Permalink:

http://dlmf.nist.gov/14.5.E6

Encodings:

pMML, png, TeX

See also:

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

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14.5.9 $${𝑸}_0\left(x\right)=\frac{1}{2}\ln \left(\frac{x+1}{x-1}\right),$$

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Symbols:

$\ln z$: principal branch of logarithm function, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $x$: real variable

A&S Ref:

8.4.2 (modified)

Permalink:

http://dlmf.nist.gov/14.5.E9

Encodings:

pMML, png, TeX

See also:

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

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14.5.10 $${𝑸}_1\left(x\right)=\frac{x}{2}\ln \left(\frac{x+1}{x-1}\right)-1.$$

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Symbols:

$\ln z$: principal branch of logarithm function, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $x$: real variable

A&S Ref:

8.4.4 (modified)

Permalink:

http://dlmf.nist.gov/14.5.E10

Encodings:

pMML, png, TeX

See also:

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

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14.5.30 $${𝑸}_2\left(x\right)=\frac{3x^2-1}{8}\ln \left(\frac{x+1}{x-1}\right)-\frac{3}{4}x.$$

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Symbols:

$\ln z$: principal branch of logarithm function, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $x$: real variable

Referenced by:

(14.5.30)

Permalink:

http://dlmf.nist.gov/14.5.E30

Encodings:

pMML, png, TeX

Addition (effective with 1.0.7):

(14.5.30) has been added to this section.

See also:

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

10: 14.12 Integral Representations

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14.12.5 $$P_{\nu }^{-\mu }\left(x\right)=\frac{{\left(x^2-1\right)}^{-\mu /2}}{\mathrm{\Gamma }\left(\mu \right)}{\int }_1^x{P}_{\nu }\left(t\right){(x-t)}^{\mu -1}dt,$$ $\mathrm{\Re }\mu >0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $P_n\left(x\right)$: Legendre polynomial, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.17(i)

Permalink:

http://dlmf.nist.gov/14.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(ii), §14.12 and Ch.14

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14.12.11 $${𝑸}_n^m\left(x\right)=\frac{{\left(x^2-1\right)}^{m/2}}{2^{n+1}n!}{\int }_{-1}^1\frac{{\left(1-t^2\right)}^n}{{(x-t)}^{n+m+1}}dt,$$

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Symbols:

${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $x$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/14.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(ii), §14.12 and Ch.14

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14.12.12 $${𝑸}_n^m\left(x\right)=\frac{1}{(n-m)!}{P}_n^m\left(x\right){\int }_x^{\mathrm{\infty }}\frac{dt}{\left(t^2-1\right){\left(P_n^m\left(t\right)\right)}^2},$$ $n\ge m$.

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $x$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§14.17(i)

Permalink:

http://dlmf.nist.gov/14.12.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(ii), §14.12 and Ch.14

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14.12.13 $${𝑸}_n\left(x\right)=\frac{1}{2(n!)}{\int }_{-1}^1\frac{P_n\left(t\right)}{x-t}dt.$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function, $x$: real variable and $n$: nonnegative integer

A&S Ref:

8.8.3 (modified)

Permalink:

http://dlmf.nist.gov/14.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(ii), §14.12(ii), §14.12 and Ch.14

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14.12.14 $${𝑸}_n\left(x\right)=\frac{1}{n!}{\int }_0^{\mathrm{\infty }}\frac{dt}{{\left(x+{(x^2-1)}^{1/2}\cosh t\right)}^{n+1}}.$$

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Symbols:

$dx$: differential of $x$ , $!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function, $\int$: integral, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function, $x$: real variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/14.12.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(ii), §14.12(ii), §14.12 and Ch.14

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