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)
A survey of numerical methods for the computation of Bessel function integrals. (English) Zbl 0598.65008
Rend. Sem. Mat., Torino, Fasc. Spec., Special functions: theory and computation, Int. Conf., Turin 1984, 249-265 (1985).
[For the entire collection see Zbl 0585.00004.] The authors discuss methods for computing numerically the integral $I(a,p,\nu)=\int\sp{a}\sb{0}J\sb{\nu}(px)f(x)dx$ where a is positive real, $J\sb{\nu}(\cdot)$ is the Bessel function of real order $\nu$, and p is a positive parameter. These integrals are difficult to compute if ap is large, if a is infinite, or if f(x) has a singularity or is oscillatory. Two methods are described to deal with the case where a is finite. One method subdivides the interval [0,a], integrating separately between the zeros $j\sb{\nu,n}$ of $J\sb{\nu}(x)$, and using the Euler transformation to speed up the convergence of the resulting series. The other method is based on a modified Clenshaw-Curtis quadrature formula. Considering the integral $I=\int\sp{1}\sb{0}x\sp{\alpha}J\sb{\nu}(\omega x)g(x)dx$ where $\alpha$ is real and such that the integral converges and g(x) is smooth, g(x) is replaced by an expansion in the shifted Chebyshev polynomials $T\sp*\sb k(x)$. This requires the calculation of the modified moments $M\sb k(\omega,x,\alpha)=\int\sp{1}\sb{0}x\sp{\alpha}J\sb{\nu}(\omega x)T\sp*\sb k(x)dx.$ These moments satisfy a homogeneous linear 9-term recurrence relation. Ways to obtain starting values for this relation are indicated and its stability is analysed. It turns out that $M\sb k(\omega,x,\alpha)$ can be computed accurately using forward recursion provided $k<<\omega /2$. Two examples are given and the results are compared with other methods. In the case where the interval of intergration is infinite, four methods are briefly described. These methods, fairly wellknown to numerical analysts, consist of (i) integrating between the zeros of $J\sb{\nu}(x)$ with convergence acceleration for the resulting series, (ii) transforming the integral into a double integral by using a suitable integral representation of $J\sb{\nu}(x)$, (iii) replacing f(x) by an expansion in suitable functions and (iv), treating the tail of the integral by asymptotic methods. Howevever, it seems doubtful whether method (ii) is in fact advantageous unless analytic integration can be used to avoid numerical evaluation of the double integral. The paper closes by announcing a project for the construction of numerical software for the computation of the modified moments for 19 different weight functions.
Reviewer: K.S.Kölbig

65D20Computation of special functions, construction of tables
65D32Quadrature and cubature formulas (numerical methods)
33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$