zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Improved calculation of prolate spheroidal radial functions of the second kind and their first derivatives. (English) Zbl 1066.65029

Summary: Alternative expressions for calculating the prolate spheroidal radial functions of the second kind R ml (2) (c,ξ) and their first derivatives with respect to ξ are shown to provide accurate values over wide parameter ranges where the traditional expressions fail to do so.

The first alternative expression is obtained from the expansion of the product of R ml (2) (c,ξ) and the prolate spheroidal angular function of the first kind S ml (1) (c,η) in a series of products of the corresponding spherical functions. A similar expression for the radial functions of the first kind was shown previously to provide accurate values for the prolate spheroidal radial functions of the first kind and their first derivatives over all parameter ranges.

The second alternative expression for R ml (2) (c,ξ) involves an integral of the product of S ml (1) (c,η) and a spherical Neumann function kernel. It provides accurate values when ξ is near unity and l-m is not too large, even when c becomes large and traditional expressions fail. The improvement in accuracy using the alternative expressions is quantified and discussed.

MSC:
65D20Computation of special functions, construction of tables
33E10Lamé, Mathieu, and spheroidal wave functions
33F05Numerical approximation and evaluation of special functions