§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 $(x,y,z)$ 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 $c$ 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, $e$ : 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), Version 1.0.1 (June 27, 2011)
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, $e$ : 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, $dx$ : differential of $x$ , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : 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
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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, $dx$ : differential of $x$ , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : 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
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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, $n$ : 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, $m$ : nonnegative integer, $n$ : 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, $m$ : nonnegative integer, $n$ : nonnegative integer and $\xi>0$ : variable
Permalink:: http://dlmf.nist.gov/14.19.E8
Encodings:: TeX, pMML, png

§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