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1: 14.19 Toroidal (or Ring) Functions

§14.19 Toroidal (or Ring) Functions

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§14.19(i) Introduction

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§14.19(ii) Hypergeometric Representations

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§14.19(iv) Sums

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

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2: 14.31 Other Applications

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§14.31(i) Toroidal Functions

►Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)). ►

§14.31(ii) Conical Functions

►The conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^m\left(x\right)$ appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). …

3: 14.34 Software

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§14.34(iv) Conical (Mehler) and/or Toroidal Functions

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4: Bibliography G

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Evaluation of toroidal harmonics. Comput. Phys. Comm. 124 (1), pp. 104–122.

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External Links:

Document, Code, ZentralBlatt

Referenced by:

4th item.

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BibTeX

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DTORH3 2.0: A new version of a computer program for the evaluation of toroidal harmonics. Comput. Phys. Comm. 139 (2), pp. 186–191.

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External Links:

Document, Code, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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Computing toroidal functions for wide ranges of the parameters. J. Comput. Phys. 161 (1), pp. 204–217.

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External Links:

ISSN 0021-9991, Document, MathReview (Adam Marlewski), ZentralBlatt

Referenced by:

§14.14, §14.15(i), §14.26, §14.31(i), 3rd item.

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BibTeX

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5: 14.1 Special Notation

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6: Bibliography V

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Expansion of vacuum magnetic fields in toroidal harmonics. Comput. Phys. Comm. 81 (1-2), pp. 74–90.

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External Links:

Document

Referenced by:

§14.31(i).

Encodings:

BibTeX

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7: Bibliography T

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Asymptotic solution of a linear nonhomogeneous second order differential equation with a transition point and its application to the computations of toroidal shells and propeller blades. J. Appl. Math. Mech. 23, pp. 1549–1565.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§9.1, Notations E, Notations E.

Encodings:

BibTeX

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