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)
Numerical calculation of singular integrals related to Hankel transform. (English) Zbl 0718.65010

The singular integral S= 0 f(x)e -x J 0 (ωx)dx is calculated numerically (J i is the Bessel function of order i, i=0,1) by using an integral expression for J 0 . If f(x) is bounded and analytic in some complex domain, the double integral obtained in this way is calculated for

|ω|1·5 by Gauss-Laguerre and Gauss-Chebyshev formulae;

|ω|>1·5 by Gauss-Laguerre formulae, changes of variables,

and Gauss-Legendre formulae. The bound 1.5 is searched by trial. Further the singular integral S ' = 0 f(x)e -x J 1 (ωx)dx is derived from S. It is stated that the FORTRAN subroutines run very fast and give a relative precision better than 5×10 -6 (for all ω).


MSC:
65D20Computation of special functions, construction of tables
65D32Quadrature and cubature formulas (numerical methods)
65R10Integral transforms (numerical methods)
33E30Functions coming from differential, difference and integral equations