Ferrers functions

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1: 14.1 Special Notation

§14.1 Special Notation

… ►The main functions treated in this chapter are the Legendre functions

\mathsf{P}_{\nu}\left(x\right)

\mathsf{Q}_{\nu}\left(x\right)

P_{\nu}\left(z\right)

Q_{\nu}\left(z\right)

; Ferrers functions

\mathsf{P}^{\mu}_{\nu}\left(x\right)

\mathsf{Q}^{\mu}_{\nu}\left(x\right)

(also known as the Legendre functions on the cut); associated Legendre functions

P^{\mu}_{\nu}\left(z\right)

Q^{\mu}_{\nu}\left(z\right)

\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)

; conical functions

\mathsf{P}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)

\mathsf{Q}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)

\widehat{\mathsf{Q}}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)

P^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)

Q^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)

(also known as Mehler functions). …

2: 14.32 Methods of Computation

§14.32 Methods of Computation

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3: 14.7 Integer Degree and Order

§14.7 Integer Degree and Order

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§14.7(ii) Rodrigues-Type Formulas

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§14.7(iii) Reflection Formulas

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§14.7(iv) Generating Functions

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4: 14.17 Integrals

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§14.17(i) Indefinite Integrals

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§14.17(ii) Barnes’ Integral

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§14.17(iii) Orthogonality Properties

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§14.17(v) Laplace Transforms

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§14.17(vi) Mellin Transforms

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5: 14.5 Special Values

§14.5 Special Values

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14.5.7

\mathsf{Q}_{0}\left(x\right)=\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right),

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Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind and $x$ : real variable
A&S Ref:: 8.4.2
Permalink:: http://dlmf.nist.gov/14.5.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §14.5(ii), §14.5 and Ch.14

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14.5.8

\mathsf{Q}_{1}\left(x\right)=\frac{x}{2}\ln\left(\frac{1+x}{1-x}\right)-1.

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Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind and $x$ : real variable
A&S Ref:: 8.4.4
Permalink:: http://dlmf.nist.gov/14.5.E8
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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6: 14.10 Recurrence Relations and Derivatives

§14.10 Recurrence Relations and Derivatives

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14.10.1

{\mathsf{P}^{\mu+2}_{\nu}\left(x\right)+2(\mu+1)x\left(1-x^{2}\right)^{-1/2}% \mathsf{P}^{\mu+1}_{\nu}\left(x\right)}+(\nu-\mu)(\nu+\mu+1)\mathsf{P}^{\mu}_{% \nu}\left(x\right)=0,

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.10
Permalink:: http://dlmf.nist.gov/14.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §14.10 and Ch.14

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14.10.2

{\left(1-x^{2}\right)^{1/2}\mathsf{P}^{\mu+1}_{\nu}\left(x\right)-(\nu-\mu+1)% \mathsf{P}^{\mu}_{\nu+1}\left(x\right)}+(\nu+\mu+1)x\mathsf{P}^{\mu}_{\nu}% \left(x\right)=0,

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.10.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §14.10 and Ch.14

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14.10.3

{(\nu-\mu+2)\mathsf{P}^{\mu}_{\nu+2}\left(x\right)-(2\nu+3)x\mathsf{P}^{\mu}_{% \nu+1}\left(x\right)}+(\nu+\mu+1)\mathsf{P}^{\mu}_{\nu}\left(x\right)=0,

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
A&S Ref:: 8.5.3 (modified)
Referenced by:: §14.10, §14.14, §14.21(iii)
Permalink:: http://dlmf.nist.gov/14.10.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §14.10 and Ch.14

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14.10.4

\left(1-x^{2}\right)\frac{\mathrm{d}\mathsf{P}^{\mu}_{\nu}\left(x\right)}{% \mathrm{d}x}={(\mu-\nu-1)\mathsf{P}^{\mu}_{\nu+1}\left(x\right)+(\nu+1)x% \mathsf{P}^{\mu}_{\nu}\left(x\right)},

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.5(i)
Permalink:: http://dlmf.nist.gov/14.10.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §14.10 and Ch.14

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7: 14.4 Graphics

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§14.4(i) Ferrers Functions: 2D Graphs

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§14.4(ii) Ferrers Functions: 3D Surfaces

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See accompanying text — Figure 14.4.15: $\mathsf{P}^{-\mu}_{0}\left(x\right)$ , $0\leq\mu\leq 10,-1<x<1$ . Magnify 3D Help

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8: 14.18 Sums

§14.18 Sums

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§14.18(ii) Addition Theorems

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14.18.3

\mathsf{Q}_{\nu}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2% }\cos\phi\right)=\mathsf{P}_{\nu}\left(\cos\theta_{1}\right)\mathsf{Q}_{\nu}% \left(\cos\theta_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}\mathsf{P}^{-m}_{\nu}% \left(\cos\theta_{1}\right)\mathsf{Q}^{m}_{\nu}\left(\cos\theta_{2}\right)\cos% \left(m\phi\right).

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\cos\NVar{z}$ : cosine function, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind, $\sin\NVar{z}$ : sine function, $m$ : nonnegative integer, $\nu$ : general degree, $\theta_{1}$ , $\theta_{2}$ : real and $\phi$ : real
Referenced by:: §14.18(ii), Ch.14
Permalink:: http://dlmf.nist.gov/14.18.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §14.18(ii), §14.18 and Ch.14

… ►The formulas are also valid with the Ferrers functions as in §14.3(i) with

\mu=0

. … ►

9: 14.9 Connection Formulas

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

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14.9.3

\mathsf{P}^{-m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu-m+1\right)}{% \Gamma\left(\nu+m+1\right)}\mathsf{P}^{m}_{\nu}\left(x\right),

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $x$ : real variable, $m$ : nonnegative integer and $\nu$ : general degree
Referenced by:: §14.17(iii), §14.18(ii), §14.3(i), §14.6(ii), §14.9(i)
Permalink:: http://dlmf.nist.gov/14.9.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §14.9(i), §14.9 and Ch.14

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14.9.4

\mathsf{Q}^{-m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu-m+1\right)}{% \Gamma\left(\nu+m+1\right)}\mathsf{Q}^{m}_{\nu}\left(x\right),

\nu\neq m-1,m-2,\dots

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $x$ : real variable, $m$ : nonnegative integer and $\nu$ : general degree
Referenced by:: §14.6(ii), §14.9(i)
Permalink:: http://dlmf.nist.gov/14.9.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §14.9(i), §14.9 and Ch.14

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

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14.9.9

\frac{2}{\Gamma\left(\nu+\mu+1\right)\Gamma\left(\mu-\nu\right)}\mathsf{Q}^{% \mu}_{\nu}\left(x\right)=-\cos\left(\nu\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(% x\right)+\cos\left(\mu\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(-x\right),

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.15(i), §14.15(ii), §14.15(v)
Permalink:: http://dlmf.nist.gov/14.9.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §14.9(ii), §14.9 and Ch.14

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10: 14.6 Integer Order

§14.6 Integer Order

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14.6.1

\mathsf{P}^{m}_{\nu}\left(x\right)=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{% \mathrm{d}}^{m}\mathsf{P}_{\nu}\left(x\right)}{{\mathrm{d}x}^{m}},

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $x$ : real variable, $m$ : nonnegative integer and $\nu$ : general degree
A&S Ref:: 8.6.6
Permalink:: http://dlmf.nist.gov/14.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §14.6(i), §14.6 and Ch.14

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14.6.2

\mathsf{Q}^{m}_{\nu}\left(x\right)=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{% \mathrm{d}}^{m}\mathsf{Q}_{\nu}\left(x\right)}{{\mathrm{d}x}^{m}}.

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Symbols:: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind, $x$ : real variable, $m$ : nonnegative integer and $\nu$ : general degree
A&S Ref:: 8.6.7
Permalink:: http://dlmf.nist.gov/14.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §14.6(i), §14.6 and Ch.14

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14.6.6

\mathsf{P}^{-m}_{\nu}\left(x\right)=\left(1-x^{2}\right)^{-m/2}\int_{x}^{1}\!% \dots\!\int_{x}^{1}\mathsf{P}_{\nu}\left(x\right)\left(\!\,\mathrm{d}x\right)^% {m}.

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$ : 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 Ferrersfunction $\mathsf{P}_{\nu}\left(x\right)$ .
See also:: Annotations for §14.6(ii), §14.6 and Ch.14

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