14 Legendre and Related Functions Real Arguments 14.18 Sums 14.20 Conical (or Mehler) Functions

§14.19 Toroidal (or Ring) Functions

ⓘ

Permalink:: http://dlmf.nist.gov/14.19
See also:: Annotations for Ch.14

§14.19(i) Introduction

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Keywords:: coordinate systems, definitions, toroidal, toroidal coordinates, toroidal functions
Notes:: See Erdélyi et al. (1953a, p. 173).
Referenced by:: §1.5(ii), §14.31(ii)
Permalink:: http://dlmf.nist.gov/14.19.i
See also:: Annotations for §14.19 and Ch.14

When $\nu=n-\frac{1}{2}$ , $n=0,1,2,\dots$ , $\mu\in\mathbb{R}$ , and $x\in(1,\infty)$ solutions of (14.2.2) are known as toroidal or ring functions. This form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates $(\eta,\theta,\phi)$ , which are related to Cartesian coordinates $(x,y,z)$ by

14.19.1	$\displaystyle x$	$\displaystyle=\frac{c\sinh\eta\cos\phi}{\cosh\eta-\cos\theta},$
	$\displaystyle y$	$\displaystyle=\frac{c\sinh\eta\sin\phi}{\cosh\eta-\cos\theta},$
	$\displaystyle z$	$\displaystyle=\frac{c\sin\theta}{\cosh\eta-\cos\theta},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\sin\NVar{z}$ : sine function, $x$ : Cartesian coordinate, $y$ : Cartesian coordinate, $z$ : Cartesian coordinate, $\eta$ : toroidal coordinate, $\theta$ : toroidal coordinate and $\phi$ : toroidal coordinate Permalink: http://dlmf.nist.gov/14.19.E1 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §14.19(i), §14.19 and Ch.14

where the constant $c$ is a scaling factor. Most required properties of toroidal functions come directly from the results for $P^{\mu}_{\nu}\left(x\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ . In particular, for $\mu=0$ and $\nu=\pm\frac{1}{2}$ see §14.5(v).

§14.19(ii) Hypergeometric Representations

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Keywords:: hypergeometric representations, toroidal functions
Notes:: See Erdélyi et al. (1953a, p. 173).
Referenced by:: §14.32
Permalink:: http://dlmf.nist.gov/14.19.ii
Amendment (effective with 1.0.1):: For (14.19.2), in addition to Erdélyi et al. (1953a, p. 173), use (5.5.5) with $z=\frac{1}{2}-\mu$ .
See also:: Annotations for §14.19 and Ch.14

With $\mathbf{F}$ as in §14.3 and $\xi>0$ ,

14.19.2

P^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(\frac{1}{2}-% \mu\right)}{\pi^{1/2}\left(1-e^{-2\xi}\right)^{\mu}e^{(\nu+(1/2))\xi}}\*% \mathbf{F}\left(\tfrac{1}{2}-\mu,\tfrac{1}{2}+\nu-\mu;1-2\mu;1-e^{-2\xi}\right),

\mu\neq\frac{1}{2},\frac{3}{2},\frac{5}{2},\ldots

ⓘ

Symbols:

\Gamma\left(\NVar{z}\right)

: gamma function,

P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{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\NVar{z}

: hyperbolic cosine function,

\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)

\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)

={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)

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:

TeX, pMML, png

Errata (effective with 1.0.1):

Originally the argument to

\mathbf{F}

was incorrect and the condition on

\mu

too weak:

	$P^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(1-2\mu\right)% 2^{2\mu}}{\Gamma\left(1-\mu\right)\left(1-e^{-2\xi}\right)^{\mu}e^{(\nu+(1/2))% \xi}}\*\mathbf{F}\left(\tfrac{1}{2}-\mu,\tfrac{1}{2}+\nu-\mu;1-2\mu;e^{-2\xi}% \right),$
$\mu\neq\frac{1}{2}$ .

Also, the factor multiplying

\mathbf{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

14.19.3		$\boldsymbol{Q}^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\pi^{1/2}% \left(1-e^{-2\xi}\right)^{\mu}}{e^{(\nu+(1/2))\xi}}\*\mathbf{F}\left(\mu+% \tfrac{1}{2},\nu+\mu+\tfrac{1}{2};\nu+1;e^{-2\xi}\right).$
ⓘ Symbols: $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $\mu$ : general order, $\nu$ : general degree and $\xi>0$ : variable Permalink: http://dlmf.nist.gov/14.19.E3 Encodings: TeX, pMML, png See also: Annotations for §14.19(ii), §14.19 and Ch.14

§14.19(iii) Integral Representations

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Keywords:: integral representations, toroidal functions
Notes:: See Erdélyi et al. (1953a, pp. 156–157).
Referenced by:: 3rd item
Permalink:: http://dlmf.nist.gov/14.19.iii
See also:: Annotations for §14.19 and Ch.14

With $\xi>0$ ,

14.19.4	$\displaystyle P^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)$	$\displaystyle=\frac{\Gamma\left(n+m+\frac{1}{2}\right)(\sinh\xi)^{m}}{2^{m}\pi% ^{1/2}\Gamma\left(n-m+\frac{1}{2}\right)\Gamma\left(m+\frac{1}{2}\right)}\*% \int_{0}^{\pi}\frac{(\sin\phi)^{2m}}{(\cosh\xi+\cos\phi\sinh\xi)^{n+m+(1/2)}}% \,\mathrm{d}\phi,$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\sin\NVar{z}$ : sine function, $m$ : nonnegative integer, $n$ : nonnegative integer and $\xi>0$ : variable A&S Ref: 8.11.2 Permalink: http://dlmf.nist.gov/14.19.E4 Encodings: TeX, pMML, png See also: Annotations for §14.19(iii), §14.19 and Ch.14
14.19.5	$\displaystyle\boldsymbol{Q}^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)$	$\displaystyle=\frac{\Gamma\left(n+\frac{1}{2}\right)}{\Gamma\left(n+m+\tfrac{1% }{2}\right)\Gamma\left(n-m+\frac{1}{2}\right)}\*\int_{0}^{\infty}\frac{\cosh% \left(mt\right)}{(\cosh\xi+\cosh t\sinh\xi)^{n+(1/2)}}\,\mathrm{d}t,$
$m<n+\tfrac{1}{2}$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $m$ : nonnegative integer, $n$ : nonnegative integer and $\xi>0$ : variable A&S Ref: 8.11.4 (modified and corrected) Permalink: http://dlmf.nist.gov/14.19.E5 Encodings: TeX, pMML, png See also: Annotations for §14.19(iii), §14.19 and Ch.14

§14.19(iv) Sums

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Keywords:: sums, toroidal functions
Notes:: See Erdélyi et al. (1953a, p. 166).
Permalink:: http://dlmf.nist.gov/14.19.iv
See also:: Annotations for §14.19 and Ch.14

With $\xi>0$ ,

14.19.6		$\boldsymbol{Q}^{\mu}_{-\frac{1}{2}}\left(\cosh\xi\right)+2\sum_{n=1}^{\infty}% \frac{\Gamma\left(\mu+n+\tfrac{1}{2}\right)}{\Gamma\left(\mu+\tfrac{1}{2}% \right)}\boldsymbol{Q}^{\mu}_{n-\frac{1}{2}}\left(\cosh\xi\right)\cos\left(n% \phi\right)=\dfrac{\left(\frac{1}{2}\pi\right)^{1/2}\left(\sinh\xi\right)^{\mu% }}{\left(\cosh\xi-\cos\phi\right)^{\mu+(1/2)}},$
$\Re\mu>-\tfrac{1}{2}$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\Re$ : real part, $n$ : nonnegative integer, $\mu$ : general order and $\xi>0$ : variable Permalink: http://dlmf.nist.gov/14.19.E6 Encodings: TeX, pMML, png See also: Annotations for §14.19(iv), §14.19 and Ch.14

§14.19(v) Whipple’s Formula for Toroidal Functions

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Keywords:: Whipple’s formula, toroidal functions
Notes:: Combine (14.9.16), (14.9.17) with (14.9.11)–(14.9.13).
Permalink:: http://dlmf.nist.gov/14.19.v
See also:: Annotations for §14.19 and Ch.14

With $\xi>0$ ,

14.19.7		$P^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(n+m+\tfrac{1}{2}% \right)}{\Gamma\left(n-m+\tfrac{1}{2}\right)}\*\left(\frac{2}{\pi\sinh\xi}% \right)^{1/2}\boldsymbol{Q}^{n}_{m-\frac{1}{2}}\left(\coth\xi\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cosh\NVar{z}$ : hyperbolic cosine function, $\coth\NVar{z}$ : hyperbolic cotangent function, $\sinh\NVar{z}$ : hyperbolic sine function, $m$ : nonnegative integer, $n$ : nonnegative integer and $\xi>0$ : variable Permalink: http://dlmf.nist.gov/14.19.E7 Encodings: TeX, pMML, png See also: Annotations for §14.19(v), §14.19 and Ch.14

14.19.8		$\boldsymbol{Q}^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(m-n+% \tfrac{1}{2}\right)}{\Gamma\left(m+n+\tfrac{1}{2}\right)}\*\left(\frac{\pi}{2% \sinh\xi}\right)^{1/2}P^{n}_{m-\frac{1}{2}}\left(\coth\xi\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cosh\NVar{z}$ : hyperbolic cosine function, $\coth\NVar{z}$ : hyperbolic cotangent function, $\sinh\NVar{z}$ : hyperbolic sine function, $m$ : nonnegative integer, $n$ : nonnegative integer and $\xi>0$ : variable Permalink: http://dlmf.nist.gov/14.19.E8 Encodings: TeX, pMML, png See also: Annotations for §14.19(v), §14.19 and Ch.14

14.18 Sums 14.20 Conical (or Mehler) Functions

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§14.19 Toroidal (or Ring) Functions

Contents

§14.19(i) Introduction

§14.19(ii) Hypergeometric Representations

§14.19(iii) Integral Representations

§14.19(iv) Sums

§14.19(v) Whipple’s Formula for Toroidal Functions

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