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

§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

§14.19(i) Introduction

Notes:: See Erdélyi et al. (1953a, p. 173).
Keywords:: coordinate systems, toroidal coordinates, toroidal functions
Referenced by:: ¶ ‣ §1.5(ii), §14.31(ii)
Permalink:: http://dlmf.nist.gov/14.19.i

When $\nu=n-\frac{1}{2}$ , $n=0,1,2,\dots$ , $\mu\in\Real$ , 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 by

14.19.1

$x=\frac{c\mathop{\sinh\/}\nolimits\eta\mathop{\cos\/}\nolimits\phi}{\mathop{% \cosh\/}\nolimits\eta-\mathop{\cos\/}\nolimits\theta},$

$y=\frac{c\mathop{\sinh\/}\nolimits\eta\mathop{\sin\/}\nolimits\phi}{\mathop{% \cosh\/}\nolimits\eta-\mathop{\cos\/}\nolimits\theta},$

$z=\frac{c\mathop{\sin\/}\nolimits\theta}{\mathop{\cosh\/}\nolimits\eta-\mathop% {\cos\/}\nolimits\theta},$

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

§14.19(ii) Hypergeometric Representations

Notes:: See Erdélyi et al. (1953a, p. 173).
Keywords:: toroidal functions
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 $z=\frac{1}{2}-\mu$ .
Reported 2010-12-10

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

14.19.2 $\mathop{P^{{\mu}}_{{\nu-\frac{1}{2}}}\/}\nolimits\!\left(\mathop{\cosh\/}% \nolimits\xi\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\mu% \right)}{\pi^{{1/2}}\left(1-e^{{-2\xi}}\right)^{\mu}e^{{(\nu+(1/2))\xi}}}\*% \mathop{\mathbf{F}\/}\nolimits\!\left(\tfrac{1}{2}-\mu,\tfrac{1}{2}+\nu-\mu;1-% 2\mu;1-e^{{-2\xi}}\right),$ $\mu\neq\frac{1}{2},\frac{3}{2},\frac{5}{2},\ldots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, $\mu$ : general order, $\nu$ : general degree and $\xi>0$ : variable
Referenced by:: §14.19(ii), 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 $\mathop{\mathbf{F}\/}\nolimits$ was incorrect and the condition on $\mu$ too weak:
$\mathop{P^{{\mu}}_{{\nu-\frac{1}{2}}}\/}\nolimits\!\left(\mathop{\cosh\/}% \nolimits\xi\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(1-2\mu\right)2^{{2% \mu}}}{\mathop{\Gamma\/}\nolimits\!\left(1-\mu\right)\left(1-e^{{-2\xi}}\right% )^{\mu}e^{{(\nu+(1/2))\xi}}}\*\mathop{\mathbf{F}\/}\nolimits\!\left(\tfrac{1}{% 2}-\mu,\tfrac{1}{2}+\nu-\mu;1-2\mu;e^{{-2\xi}}\right),$ $\mu\neq\frac{1}{2}$ .
Also, the factor multiplying $\mathop{\mathbf{F}\/}\nolimits$ was rewritten to clarify the poles, which determine the correct condition on $\mu$ .
Reported 2010-11-02 by Alvaro Valenzuela

14.19.3 $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu-\frac{1}{2}}}\/}\nolimits\!\left(\mathop{% \cosh\/}\nolimits\xi\right)=\frac{\pi^{{1/2}}\left(1-e^{{-2\xi}}\right)^{\mu}}% {e^{{(\nu+(1/2))\xi}}}\*\mathop{\mathbf{F}\/}\nolimits\!\left(\mu+\tfrac{1}{2}% ,\nu+\mu+\tfrac{1}{2};\nu+1;e^{{-2\xi}}\right).$

Symbols:: $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : 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

§14.19(iii) Integral Representations

Notes:: See Erdélyi et al. (1953a, pp. 156–157).
Keywords:: toroidal functions
Referenced by:: §14.32
Permalink:: http://dlmf.nist.gov/14.19.iii

With $\xi>0$ ,

14.19.4 $\mathop{P^{{m}}_{{n-\frac{1}{2}}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits% \xi\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(n+m+\frac{1}{2}\right)(% \mathop{\sinh\/}\nolimits\xi)^{m}}{2^{m}\pi^{{1/2}}\mathop{\Gamma\/}\nolimits% \!\left(n-m+\frac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(m+\frac{1}{2}% \right)}\*\int_{{0}}^{{\pi}}\frac{(\mathop{\sin\/}\nolimits\phi)^{{2m}}}{(% \mathop{\cosh\/}\nolimits\xi+\mathop{\cos\/}\nolimits\phi\mathop{\sinh\/}% \nolimits\xi)^{{n+m+(1/2)}}}d\phi,$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : nonnegative integer, : nonnegative integer and $\xi>0$ : variable
A&S Ref:: 8.11.2
Permalink:: http://dlmf.nist.gov/14.19.E4
Encodings:: TeX, pMML, png

14.19.5 $\mathop{\boldsymbol{Q}^{{m}}_{{n-\frac{1}{2}}}\/}\nolimits\!\left(\mathop{% \cosh\/}\nolimits\xi\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(n+\frac{1}% {2}\right)}{\mathop{\Gamma\/}\nolimits\!\left(n+m+\tfrac{1}{2}\right)\mathop{% \Gamma\/}\nolimits\!\left(n-m+\frac{1}{2}\right)}\*\int_{{0}}^{{\infty}}\frac{% \mathop{\cosh\/}\nolimits\!\left(mt\right)}{(\mathop{\cosh\/}\nolimits\xi+% \mathop{\cosh\/}\nolimits t\mathop{\sinh\/}\nolimits\xi)^{{n+(1/2)}}}dt,$ $m<n+\tfrac{1}{2}$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : differential of , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, : nonnegative integer, : nonnegative integer and $\xi>0$ : variable
A&S Ref:: 8.11.4 (modified and corrected)
Permalink:: http://dlmf.nist.gov/14.19.E5
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§14.19(iv) Sums

Notes:: See Erdélyi et al. (1953a, p. 166).
Keywords:: toroidal functions
Permalink:: http://dlmf.nist.gov/14.19.iv

With $\xi>0$ ,

14.19.6 $\mathop{\boldsymbol{Q}^{{\mu}}_{{-\frac{1}{2}}}\/}\nolimits\!\left(\mathop{% \cosh\/}\nolimits\xi\right)+2\sum_{{n=1}}^{{\infty}}\frac{\mathop{\Gamma\/}% \nolimits\!\left(\mu+n+\tfrac{1}{2}\right)}{\mathop{\Gamma\/}\nolimits\!\left(% \mu+\tfrac{1}{2}\right)}\mathop{\boldsymbol{Q}^{{\mu}}_{{n-\frac{1}{2}}}\/}% \nolimits\!\left(\mathop{\cosh\/}\nolimits\xi\right)\mathop{\cos\/}\nolimits\!% \left(n\phi\right)=\dfrac{\left(\frac{1}{2}\pi\right)^{{1/2}}\left(\mathop{% \sinh\/}\nolimits\xi\right)^{\mu}}{\left(\mathop{\cosh\/}\nolimits\xi-\mathop{% \cos\/}\nolimits\phi\right)^{{\mu+(1/2)}}},$ $\realpart{\mu}>-\tfrac{1}{2}$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\realpart{}$ : real part, : nonnegative integer, $\mu$ : general order and $\xi>0$ : variable
Permalink:: http://dlmf.nist.gov/14.19.E6
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§14.19(v) Whipple’s Formula for Toroidal Functions

Notes:: Combine (14.9.16), (14.9.17) with (14.9.11)–(14.9.13).
Keywords:: Whipple’s formula, toroidal functions
Permalink:: http://dlmf.nist.gov/14.19.v

With $\xi>0$ ,

14.19.7 $\mathop{P^{{m}}_{{n-\frac{1}{2}}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits% \xi\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(n+m+\tfrac{1}{2}\right)}{% \mathop{\Gamma\/}\nolimits\!\left(n-m+\tfrac{1}{2}\right)}\*\left(\frac{2}{\pi% \mathop{\sinh\/}\nolimits\xi}\right)^{{1/2}}\mathop{\boldsymbol{Q}^{{n}}_{{m-% \frac{1}{2}}}\/}\nolimits\!\left(\mathop{\coth\/}\nolimits\xi\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, : nonnegative integer, : nonnegative integer and $\xi>0$ : variable
Permalink:: http://dlmf.nist.gov/14.19.E7
Encodings:: TeX, pMML, png

14.19.8 $\mathop{\boldsymbol{Q}^{{m}}_{{n-\frac{1}{2}}}\/}\nolimits\!\left(\mathop{% \cosh\/}\nolimits\xi\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(m-n+\tfrac% {1}{2}\right)}{\mathop{\Gamma\/}\nolimits\!\left(m+n+\tfrac{1}{2}\right)}\*% \left(\frac{\pi}{2\mathop{\sinh\/}\nolimits\xi}\right)^{{1/2}}\mathop{P^{{n}}_% {{m-\frac{1}{2}}}\/}\nolimits\!\left(\mathop{\coth\/}\nolimits\xi\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, : nonnegative integer, : nonnegative integer and $\xi>0$ : variable
Permalink:: http://dlmf.nist.gov/14.19.E8
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 14.18 Sums 14.20 Conical (or Mehler) Functions

§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