Olver associated Legendre function

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11: 14.22 Graphics

§14.22 Graphics

►In the graphics shown in this section, height corresponds to the absolute value of the function and color to the phase. … ►

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Figure 14.22.1: $P_{1/2}^0\left(x+\mathrm{i}y\right)$, $-5\le x\le 5$, $-5\le y\le 5$. … Magnify 3D Help

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Figure 14.22.4: ${𝑸}_0^0\left(x+\mathrm{i}y\right)$, $-5\le x\le 5$, $-5\le y\le 5$. … Magnify 3D Help

12: 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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14.5.17 $${𝑸}_{\nu }^{\pm 1/2}\left(\cosh \xi \right)={\left(\frac{\pi }{2\sinh \xi }\right)}^{1/2}\frac{\exp \left(-\left(\nu +\frac{1}{2}\right)\xi \right)}{\mathrm{\Gamma }\left(\nu +\frac{3}{2}\right)}.$$

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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, $\exp z$: exponential function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\nu$: general degree and $\xi >0$: variable

A&S Ref:

8.6.10 (corrected) 8.6.11 (corrected)

Permalink:

http://dlmf.nist.gov/14.5.E17

Encodings:

pMML, png, TeX

See also:

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

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14.5.27 $${𝑸}_{-\frac{1}{2}}\left(\cosh \xi \right)=2{\pi }^{-1/2}{\mathrm{e}}^{-\xi /2}K\left({\mathrm{e}}^{-\xi }\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $\xi >0$: variable

A&S Ref:

8.13.4 (modified)

Permalink:

http://dlmf.nist.gov/14.5.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(v), §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

13: 14.6 Integer Order

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14.6.5 $${\left(\nu +1\right)}_m{𝑸}_{\nu }^m\left(x\right)={(-1)}^m{\left(x^2-1\right)}^{m/2}\frac{d^m{𝑸}_{\nu }\left(x\right)}{{dx}^m}.$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

Referenced by:

§14.21(iii), §14.6(i)

Permalink:

http://dlmf.nist.gov/14.6.E5

Encodings:

pMML, png, TeX

See also:

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

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14: 14.2 Differential Equations

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

14.2.8 $$𝒲\left\{P_{\nu }^{-\mu }\left(x\right),{𝑸}_{\nu }^{\mu }\left(x\right)\right\}=-\frac{1}{\mathrm{\Gamma }\left(\nu +\mu +1\right)\left(x^2-1\right)},$$

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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, $𝒲$: Wronskian, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $x$: real variable, $\mu$: general order and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.2.E8

Encodings:

pMML, png, TeX

See also:

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

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14.2.9 $$𝒲\left\{{𝑸}_{\nu }^{\mu }\left(x\right),{𝑸}_{-\nu -1}^{\mu }\left(x\right)\right\}=\frac{\cos \left(\nu \pi \right)}{x^2-1},$$

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Symbols:

$𝒲$: Wronskian, ${𝑸}_{\nu }^{\mu }\left(z\right)$: 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

Permalink:

http://dlmf.nist.gov/14.2.E9

Encodings:

pMML, png, TeX

See also:

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

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15: 14.7 Integer Degree and Order

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14.7.6 $$Q_n^0\left(x\right)=Q_n\left(x\right)=n!{𝑸}_n^0\left(x\right)=n!{𝑸}_n\left(x\right),$$

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Symbols:

$Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $!$: factorial (as in $n!$), $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function, $x$: real variable and $n$: nonnegative integer

Referenced by:

§14.21(iii)

Permalink:

http://dlmf.nist.gov/14.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §14.7(i), §14.7 and Ch.14

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16: 14.15 Uniform Asymptotic Approximations

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14.15.3 $${𝑸}_{\nu }^{\mu }\left(x\right)=\frac{1}{{\mu }^{\nu +(1/2)}}{\left(\frac{\pi u}{2}\right)}^{1/2}{I}_{\nu +\frac{1}{2}}\left(\mu u\right)\left(1+O\left(\frac{1}{\mu }\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $x$: real variable, $\mu$: general order, $\nu$: general degree and $u$

Referenced by:

§14.15(i)

Permalink:

http://dlmf.nist.gov/14.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(i), §14.15 and Ch.14

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14.15.9 $${𝑸}_{\nu }^{\mu }\left(x\right)={\left(\frac{\pi }{2}\right)}^{1/2}{\left(\frac{\mathrm{e}}{\mu }\right)}^{\nu +(1/2)}{\left(\frac{1-\alpha }{1+\alpha }\right)}^{\mu /2}{\left(1-{\alpha }^2\right)}^{-(\nu /2)-(1/4)}{\left(\frac{{\alpha }^2+{\eta }^2}{{\alpha }^2\left(x^2-1\right)+1}\right)}^{1/4}{I}_{\nu +\frac{1}{2}}\left(\mu \eta \right)\left(1+O\left(\frac{1}{\mu }\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $x$: real variable, $\mu$: general order, $\nu$: general degree, $\alpha$ and $\eta$

Permalink:

http://dlmf.nist.gov/14.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(ii), §14.15 and Ch.14

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14.15.14 $${𝑸}_{\nu }^{\mu }\left(\cosh \xi \right)=\frac{{\nu }^{\mu }}{\mathrm{\Gamma }\left(\nu +\mu +1\right)}{\left(\frac{\xi }{\sinh \xi }\right)}^{1/2}{K}_{\mu }\left(\left(\nu +\frac{1}{2}\right)\xi \right)\left(1+O\left(\frac{1}{\nu }\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mu$: general order, $\nu$: general degree and $\xi$: variable

Permalink:

http://dlmf.nist.gov/14.15.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(iii), §14.15 and Ch.14

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14.15.18 $${𝑸}_{\nu }^{\mu }\left(x\right)=\frac{1}{\beta \mathrm{\Gamma }\left(\nu +\mu +1\right)}{\left(\frac{{\alpha }^2-y}{x^2-1+{\alpha }^2}\right)}^{1/4}{K}_{\mu }\left(\left(\nu +\frac{1}{2}\right){|y|}^{1/2}\right)\left(1+O\left(\frac{1}{\nu }\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $x$: real variable, $\mu$: general order, $\nu$: general degree, $\beta$ , $y$ and $\alpha$

Referenced by:

§14.15(iv)

Permalink:

http://dlmf.nist.gov/14.15.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(iv), §14.15 and Ch.14

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17: Bibliography O

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Associated Legendre functions on the cut. J. Comput. Phys. 51 (3), pp. 502–518.

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Notes:

Includes single, double precision Fortran code.

External Links:

GAMS (module DXNRMP; package SLATEC), ISSN 0021-9991, Document, MathReview (G. P. Bhattacharjee), ZentralBlatt

Referenced by:

§14.32, 7th item.

Encodings:

BibTeX

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18: 14.28 Sums

§14.28 Sums

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§14.28(i) Addition Theorem

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14.28.1 $$P_{\nu }\left(z_1{z}_2-{\left(z_1^2-1\right)}^{1/2}{\left(z_2^2-1\right)}^{1/2}\cos \varphi \right)=P_{\nu }\left(z_1\right)P_{\nu }\left(z_2\right)+2\sum _{m=1}^{\mathrm{\infty }}{(-1)}^m\frac{\mathrm{\Gamma }\left(\nu -m+1\right)}{\mathrm{\Gamma }\left(\nu +m+1\right)}{P}_{\nu }^m\left(z_1\right)P_{\nu }^m(z_2)\cos \left(m\varphi \right),$$

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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, $\cos z$: cosine function, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $z$: complex variable, $m$: nonnegative integer and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.28.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.28(i), §14.28 and Ch.14

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§14.28(ii) Heine’s Formula

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14.28.2 $$\sum _{n=0}^{\mathrm{\infty }}(2n+1)Q_n\left(z_1\right)P_n\left(z_2\right)=\frac{1}{z_1-z_2},$$ $z_1\in {ℰ}_1$, $z_2\in {ℰ}_2$,

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Symbols:

$\in$: element of, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $z$: complex variable, $n$: nonnegative integer and ${ℰ}_1$, ${ℰ}_2$: ellipses

Permalink:

http://dlmf.nist.gov/14.28.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.28(ii), §14.28 and Ch.14

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19: 14.1 Special Notation

§14.1 Special Notation

… ►The main functions treated in this chapter are the Legendre functions ${𝖯}_{\nu }\left(x\right)$, ${𝖰}_{\nu }\left(x\right)$, $P_{\nu }\left(z\right)$, $Q_{\nu }\left(z\right)$; Ferrers functions ${𝖯}_{\nu }^{\mu }\left(x\right)$, ${𝖰}_{\nu }^{\mu }\left(x\right)$ (also known as the Legendre functions on the cut); associated Legendre functions $P_{\nu }^{\mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, ${𝑸}_{\nu }^{\mu }\left(z\right)$; conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${𝖰}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $P_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $Q_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$ (also known as Mehler functions). ►Among other notations commonly used in the literature Erdélyi et al. (1953a) and Olver (1997b) denote ${𝖯}_{\nu }^{\mu }\left(x\right)$ and ${𝖰}_{\nu }^{\mu }\left(x\right)$ by ${\mathrm{P}}_{\nu }^{\mu }(x)$ and ${\mathrm{Q}}_{\nu }^{\mu }(x)$, respectively. …Hobson (1931) denotes both ${𝖯}_{\nu }^{\mu }\left(x\right)$ and $P_{\nu }^{\mu }\left(x\right)$ by $P_{\nu }^{\mu }\left(x\right)$; similarly for ${𝖰}_{\nu }^{\mu }\left(x\right)$ and $Q_{\nu }^{\mu }\left(x\right)$.

20: 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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