associated Legendre equation

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11: 14.19 Toroidal (or Ring) Functions

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

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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, $\pi$: 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

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12: Mathematical Introduction

… ►Also, valuable initial advice on all aspects of the project was provided by ten external associate editors. … ►These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). … ►Other examples are: (a) the notation for the Ferrers functions—also known as associated Legendre functions on the cut—for which existing notations can easily be confused with those for other associated Legendre functions (§14.1); (b) the spherical Bessel functions for which existing notations are unsymmetric and inelegant (§§10.47(i) and 10.47(ii)); and (c) elliptic integrals for which both Legendre’s forms and the more recent symmetric forms are treated fully (Chapter 19). … ►For all equations and other technical information this Handbook and the DLMF either provide references to the literature for proof or describe steps that can be followed to construct a proof. … ►For equations or other technical information that appeared previously in AMS 55, the DLMF usually includes the corresponding AMS 55 equation number, or other form of reference, together with corrections, if needed. …

13: 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

… ►In November 2015, Dunster was named Associate Editor for his chapter.

14: Bibliography S

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Evaluation of associated Legendre functions off the cut and parabolic cylinder functions. Electron. Trans. Numer. Anal. 9, pp. 137–146.

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Notes:

Orthogonal polynomials: numerical and symbolic algorithms (Leganés, 1998)

External Links:

ISSN 1068-9613, FullText, MathReview (Adam Marlewski), ZentralBlatt

Referenced by:

§14.30(i), 3rd item.

Encodings:

BibTeX

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The expansion of Lamé functions into series of associated Legendre functions of the second kind. Proc. Cambridge Philos. Soc. 62, pp. 441–452.

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External Links:

Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.17(i).

Encodings:

BibTeX

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On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 46 (1), pp. 231–260.

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External Links:

ISSN 0259-9791, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

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On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 49 (7), pp. 1436–1477.

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External Links:

ISSN 0259-9791, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

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On parameter derivatives of the associated Legendre function of the first kind (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Anal. Appl. 386 (1), pp. 332–342.

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External Links:

ISSN 0022-247X, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

15: 14.8 Behavior at Singularities

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14.8.9 $${𝑸}_{\nu }\left(x\right)=-\frac{\ln \left(x-1\right)}{2\mathrm{\Gamma }\left(\nu +1\right)}+\frac{\frac{1}{2}\ln 2-\gamma -\psi \left(\nu +1\right)}{\mathrm{\Gamma }\left(\nu +1\right)}+O\left(\left(x-1\right)\ln \left(x-1\right)\right),$$ $\nu \ne -1,-2,-3,\mathrm{\dots }$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function, $x$: real variable and $\nu$: general degree

Referenced by:

Erratum (V1.1.5) for Equation (14.8.9)

Permalink:

http://dlmf.nist.gov/14.8.E9

Encodings:

pMML, png, TeX

Correction (effective with 1.1.5):

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

Suggested 2022-02-09 by Mark Ashbaugh

See also:

Annotations for §14.8(ii), §14.8 and Ch.14

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16: 10.22 Integrals

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10.22.72 $${\int }_0^{\mathrm{\infty }}{J}_{\mu }\left(at\right)J_{\nu }\left(bt\right)J_{\nu }\left(ct\right)t^{1-\mu }dt=\frac{{(bc)}^{\mu -1}\sin \left((\mu -\nu )\pi \right){(\sinh \chi )}^{\mu -\frac{1}{2}}}{{(\frac{1}{2}{\pi }^3)}^{\frac{1}{2}}{a}^{\mu }}{\mathrm{e}}^{(\mu -\frac{1}{2})\mathrm{i}\pi }{Q}_{\nu -\frac{1}{2}}^{\frac{1}{2}-\mu }\left(\cosh \chi \right),$$ $\mathrm{\Re }\mu >-\frac{1}{2},\mathrm{\Re }\nu >-1,a>b+c,\cosh \chi =(a^2-b^2-c^2)/(2bc)$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel 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, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function and $\nu$: complex parameter

Keywords:

Mellin transform

Referenced by:

§10.22(iv), §10.22(iv), Erratum (V1.0.21) for Equation (10.22.72)

Permalink:

http://dlmf.nist.gov/10.22.E72

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.21):

Originally, the factor on the right-hand side was written as $\frac{{(bc)}^{\mu -1}\cos \left(\nu \pi \right){(\sinh \chi )}^{\mu -\frac{1}{2}}}{{(\frac{1}{2}{\pi }^3)}^{\frac{1}{2}}{a}^{\mu }}$, which was taken directly from Watson (1944, p. 412, (13.46.5)), who uses a different normalization for the associatedLegendre functions of the second kind $Q_{\nu }^{\mu }$. Watson’s $Q_{\nu }^{\mu }$ equals $\frac{\sin \left((\nu +\mu )\pi \right)}{\sin \left(\nu \pi \right)}{\mathrm{e}}^{-\mu \pi \mathrm{i}}{Q}_{\nu }^{\mu }$ in the DLMF.

See also:

Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

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17: 18.30 Associated OP’s

§18.30 Associated OP’s

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§18.30(ii) Associated Legendre Polynomials

… ►For further results on associated Legendre polynomials see Chihara (1978, Chapter VI, §12). … ►

Associated Monic OP’s

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18: 10.43 Integrals

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10.43.22

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part and $\nu$: complex parameter

Keywords:

Laplace transform, Mellin transform

Referenced by:

§10.43(iv)

Permalink:

http://dlmf.nist.gov/10.43.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(iv), §10.43 and Ch.10

►For the second equation there is a cut in the $a$-plane along the interval $[0,1]$, and all quantities assume their principal values (§4.2(i)). For the Ferrers function $𝖯$ and the associated Legendre function $P$, see §§14.3(i) and 14.21(i). …

19: 14.17 Integrals

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§14.17(iii) Orthogonality Properties

… ►Orthogonality relations for the associated Legendre functions of imaginary order are given in Bielski (2013). ►

§14.17(iv) Definite Integrals of Products

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§14.17(v) Laplace Transforms

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§14.17(vi) Mellin Transforms

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20: 14.6 Integer Order

§14.6 Integer Order

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14.6.3 $$P_{\nu }^m\left(x\right)={\left(x^2-1\right)}^{m/2}\frac{d^m{P}_{\nu }\left(x\right)}{{dx}^m},$$

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

A&S Ref:

8.6.6

Referenced by:

§14.21(iii)

Permalink:

http://dlmf.nist.gov/14.6.E3

Encodings:

pMML, png, TeX

See also:

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

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14.6.4 $$Q_{\nu }^m\left(x\right)={\left(x^2-1\right)}^{m/2}\frac{d^m{Q}_{\nu }\left(x\right)}{{dx}^m},$$

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Symbols:

$Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

A&S Ref:

8.6.7

Referenced by:

§14.6(i)

Permalink:

http://dlmf.nist.gov/14.6.E4

Encodings:

pMML, png, TeX

See also:

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

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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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