zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Computing toroidal functions for wide ranges of the parameters. (English) Zbl 0994.65018
Summary: Associated Legendre functions of half-odd degree and arguments larger than one, also known as toroidal harmonics, appear in the solution of Dirichlet problems with toroidal symmetry. It is shown how the use of series expansions, continued fractions, and uniform asymptotic expansions, together with the application of recurrence relations over degrees and orders, permits the evaluation of the whole set of toroidal functions for a wide range of arguments, orders, and degrees. In particular, we provide a suitable uniform asymptotic expansion for $P^m(x)$ (for large $m$), which fills the gap left by previous methods.

65D20Computation of special functions, construction of tables
33C47Other special orthogonal polynomials and functions
Full Text: DOI
[1] M. Abramowitz and I. A. Stegun Eds., Handbook of Mathematical Functions National Bureau of Standards Applied Mathematics Series No. 55. U.S. Government Printing Office, Washington, DC.
[2] Boyd, W. G. C.; Dunster, T. M.: Uniform asymptotic solutions of a class of second-order linear differential equations having a turning point and a regular singularity, with an application to Legendre functions. SIAM J. Math. anal. 17, 422 (1986) · Zbl 0591.34048
[3] Brezinski, C.: Continued fractions and Padé approximants. 142 (1990) · Zbl 0712.00008
[4] Carlson, B. C.: Computing elliptic integrals by duplication. Numer. math. 33, 1 (1979) · Zbl 0438.65029
[5] Carlson, B. C.; Notis, E. M.: Algorithm 577: algorithms for incomplete elliptic integrals. ACM trans. Math. soft. 7, 398 (1981) · Zbl 0464.65008
[6] Dunster, T. M.: Conical functions with one or both parameters large. Proc. royal soc. Edinburgh 119A, 311 (1991) · Zbl 0736.33002
[7] Fettis, H. E.: A new method for computing toroidal harmonics. Math. comput. 24, 667 (1970) · Zbl 0215.55303
[8] Frenzen, C. L.: Error bounds for a uniform asymptotic expansion of the Legendre function qn-m (cosh z). SIAM J. Math. anal. 21, 523 (1990) · Zbl 0688.41034
[9] Gautschi, W.: Commun. ACM. 8, 488 (1965)
[10] Gautschi, W.: Computational aspects of three-term recurrence relations. SIAM rev. 9, 24 (1967) · Zbl 0168.15004
[11] Gil, A.; Segura, J.: Evaluation of Legendre functions of argument greater than one. Comput. phys. Commun. 105, 273 (1997) · Zbl 0930.65010
[12] Gil, A.; Segura, J.: A code to evaluate prolate and oblate spheroidal harmonics. Comput. phys. Commun. 108, 267 (1998) · Zbl 0937.65021
[13] Hoyles, M.; Kuyucak, S.; Chung, S. -H.: Solutions of Poisson’s equation in channel-like geometries. Comput. phys. Commun. 115, 45 (1998) · Zbl 1001.65131
[14] Kuyucak, S.; Hoyles, M.; Chung, S. -H.: Analytical solutions of Poisson’s equation for realistic geometrical shapes of membrane ion channels. Biophys. J. 74, 22 (1998) · Zbl 1001.65131
[15] Lebedev, N. N.: Special functions and their applications. (1972) · Zbl 0271.33001
[16] Van Milligen, B. Ph.; Fraguas, A. López: Expansion of vacuum magnetic fields in toroidal harmonics. Comput. phys. Commun. 81, 74 (1994)
[17] Van Milligen, B. Ph.: Exact relations between multipole moments of the flux and moments of the toroidal current density in tokamaks. Nucl. fusion 30, 157 (1990)
[18] Press, W. H.; Teukolski, S. A.; Vetterling, W. T.; Flannery, B. P.: Numerical recipes in Fortran. (1992) · Zbl 0778.65002
[19] Olver, F. W. J.; Smith, J. M.: Associated Legendre functions on the cut. J. comput. Phys. 51, 502 (1983) · Zbl 0523.65016
[20] F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974. [Reprinted, A. K. Peters, 1997, ] · Zbl 0303.41035
[21] Segura, J.; Gil, A.: Evaluation of toroidal harmonics. Comput. phys. Commun. 124, 104 (2000) · Zbl 0949.65016
[22] Temme, N. M.: Special functions: an introduction to the classical functions of mathematical physics. (1996) · Zbl 0856.33001
[23] Ursell, F.: Integrals with a large parameter: Legendre functions of large degree and fixed order. Math. proc. Camb. philos. Soc. 95, 367 (1984) · Zbl 0539.33005