Legendre functions

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11: 14.7 Integer Degree and Order

§14.7 Integer Degree and Order

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§14.7(i) $\mu =0$

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§14.7(ii) Rodrigues-Type Formulas

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§14.7(iv) Generating Functions

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12: 14.18 Sums

§14.18 Sums

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§14.18(ii) Addition Theorems

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Dougall’s Expansion

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13: 14.3 Definitions and Hypergeometric Representations

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§14.3(ii) Interval

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Associated Legendre Function of the First Kind

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Associated Legendre Function of the Second Kind

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§14.3(iv) Relations to Other Functions

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14: 14.5 Special Values

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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(v) $\mu =0$, $\nu =\pm \frac{1}{2}$

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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

15: 14.32 Methods of Computation

§14.32 Methods of Computation

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16: 14.25 Integral Representations

§14.25 Integral Representations

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14.25.1 $$P_{\nu }^{-\mu }\left(z\right)=\frac{{\left(z^2-1\right)}^{\mu /2}}{2^{\nu }\mathrm{\Gamma }\left(\mu -\nu \right)\mathrm{\Gamma }\left(\nu +1\right)}{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\nu +1}}{{(z+\cosh t)}^{\nu +\mu +1}}dt,$$ $\mathrm{\Re }\mu >\mathrm{\Re }\nu >-1$,

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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, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.25

Permalink:

http://dlmf.nist.gov/14.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.25 and Ch.14

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14.25.2 $${𝑸}_{\nu }^{\mu }\left(z\right)=\frac{{\pi }^{1/2}{\left(z^2-1\right)}^{\mu /2}}{2^{\mu }\mathrm{\Gamma }\left(\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\mu }}{{\left(z+{(z^2-1)}^{1/2}\cosh t\right)}^{\nu +\mu +1}}dt,$$ $\mathrm{\Re }\left(\nu +1\right)>\mathrm{\Re }\mu >-\frac{1}{2}$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.25

Permalink:

http://dlmf.nist.gov/14.25.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.25 and Ch.14

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17: 14.12 Integral Representations

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§14.12(ii)

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

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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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Neumann’s Integral

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Heine’s Integral

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18: 14.24 Analytic Continuation

§14.24 Analytic Continuation

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14.24.1 $$P_{\nu }^{-\mu }\left(z{\mathrm{e}}^{s\pi \mathrm{i}}\right)={\mathrm{e}}^{s\nu \pi \mathrm{i}}{P}_{\nu }^{-\mu }\left(z\right)+\frac{2\mathrm{i}\sin \left(\left(\nu +\frac{1}{2}\right)s\pi \right){\mathrm{e}}^{-s\pi \mathrm{i}/2}}{\cos \left(\nu \pi \right)\mathrm{\Gamma }\left(\mu -\nu \right)}{𝑸}_{\nu }^{\mu }\left(z\right),$$

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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, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Referenced by:

§14.24, §14.24

Permalink:

http://dlmf.nist.gov/14.24.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

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14.24.3 $$P_{\nu ,s}^{-\mu }\left(z\right)={\mathrm{e}}^{s\mu \pi \mathrm{i}}{P}_{\nu }^{-\mu }\left(z\right),$$

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: 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, $\mathrm{i}$: imaginary unit, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Permalink:

http://dlmf.nist.gov/14.24.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

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14.24.4 $${𝑸}_{\nu ,s}^{\mu }\left(z\right)={\mathrm{e}}^{-s\mu \pi \mathrm{i}}{𝑸}_{\nu }^{\mu }\left(z\right)-\frac{\pi \mathrm{i}\sin \left(s\mu \pi \right)}{\sin \left(\mu \pi \right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}{P}_{\nu }^{-\mu }\left(z\right),$$

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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, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Referenced by:

§14.24

Permalink:

http://dlmf.nist.gov/14.24.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

… ►For fixed $z$, other than $\pm 1$ or $\mathrm{\infty }$, each branch of $P_{\nu }^{-\mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ is an entire function of each parameter $\nu$ and $\mu$. …

19: 14.9 Connection Formulas

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§14.9(i) Connections Between ${𝖯}_{\nu }^{\pm \mu }\left(x\right)$, ${𝖯}_{-\nu -1}^{\pm \mu }\left(x\right)$, ${𝖰}_{\nu }^{\pm \mu }\left(x\right)$, ${𝖰}_{-\nu -1}^{\mu }\left(x\right)$

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§14.9(ii) Connections Between ${𝖯}_{\nu }^{\pm \mu }\left(\pm x\right)$, ${𝖰}_{\nu }^{-\mu }\left(\pm x\right)$, ${𝖰}_{\nu }^{\mu }\left(x\right)$

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§14.9(iii) Connections Between $P_{\nu }^{\pm \mu }\left(x\right)$, $P_{-\nu -1}^{\pm \mu }\left(x\right)$, ${𝑸}_{\nu }^{\pm \mu }\left(x\right)$, ${𝑸}_{-\nu -1}^{\mu }\left(x\right)$

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14.9.12 $$\cos \left(\nu \pi \right)P_{\nu }^{-\mu }\left(x\right)=-\frac{{𝑸}_{\nu }^{\mu }\left(x\right)}{\mathrm{\Gamma }\left(\mu -\nu \right)}+\frac{{𝑸}_{-\nu -1}^{\mu }\left(x\right)}{\mathrm{\Gamma }\left(\nu +\mu +1\right)}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.2.2 (modified)

Referenced by:

§14.20(i), §14.8(ii), §14.8(iii)

Permalink:

http://dlmf.nist.gov/14.9.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §14.9(iii), §14.9 and Ch.14

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§14.9(iv) Whipple’s Formula

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20: 14.2 Differential Equations

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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 ${𝖯}_{\nu }^0\left(x\right)={𝖯}_{\nu }\left(x\right)$, ${𝖰}_{\nu }^0\left(x\right)={𝖰}_{\nu }\left(x\right)$, $P_{\nu }^0\left(x\right)=P_{\nu }\left(x\right)$, $Q_{\nu }^0\left(x\right)=Q_{\nu }\left(x\right)$, ${𝑸}_{\nu }^0\left(x\right)={𝑸}_{\nu }\left(x\right)=Q_{\nu }\left(x\right)/\mathrm{\Gamma }\left(\nu +1\right)$. … ►Unless stated otherwise in §§14.2–14.20 it is assumed that the arguments of the functions ${𝖯}_{\nu }^{\mu }\left(x\right)$ and ${𝖰}_{\nu }^{\mu }\left(x\right)$ lie in the interval $(-1,1)$, and the arguments of the functions $P_{\nu }^{\mu }\left(x\right)$, $Q_{\nu }^{\mu }\left(x\right)$, and ${𝑸}_{\nu }^{\mu }\left(x\right)$ lie in the interval $(1,\mathrm{\infty })$. … ►

14.2.10 $$𝒲\left\{P_{\nu }^{\mu }\left(x\right),Q_{\nu }^{\mu }\left(x\right)\right\}=-{\mathrm{e}}^{\mu \pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(\nu +\mu +1\right)}{\mathrm{\Gamma }\left(\nu -\mu +1\right)\left(x^2-1\right)},$$

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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, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.1.8 (modified)

Permalink:

http://dlmf.nist.gov/14.2.E10

Encodings:

pMML, png, TeX

See also:

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

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14.2.11 $$P_{\nu +1}^{\mu }\left(x\right)Q_{\nu }^{\mu }\left(x\right)-P_{\nu }^{\mu }\left(x\right)Q_{\nu +1}^{\mu }\left(x\right)={\mathrm{e}}^{\mu \pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(\nu +\mu +1\right)}{\mathrm{\Gamma }\left(\nu -\mu +2\right)}.$$

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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, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.21(iii)

Permalink:

http://dlmf.nist.gov/14.2.E11

Encodings:

pMML, png, TeX

See also:

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