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 complex Airy functions by numerical quadrature. (English) Zbl 1012.33001
Airy functions are solutions of the differential equation $$ \frac{d^{2}w}{dz^{2}}-zw=0. $$ Two linearly independent solutions that are real for real values of $z$ are denoted by $\text{Ai}(z)$ and $\text{Bi}(z)$. They have the integral representation $$\align {\text{Ai}}(z) &=\frac{1}{\pi}\int_{0}^{\infty}\cos(zt+\frac{t^3}{3}) dt,\\ \text{Bi} (z) &=\frac{1}{\pi}\int_{0}^{\infty}\sin(zt+\frac{t^3}{3}) dt+\frac{1}{\pi}\int_{0}^{\infty}e^{zt-t^3/3} dt, \endalign$$ where we assume that $z$ is real. In this paper the authors are concerned with the numerical evaluation of $\text{Ai}(z)$ and $\text{Ai}'(z)$ for complex values of $z$ by numerical quadrature. In a first method contour integral representations of the Airy functions are written as non-oscillating integrals for obtaining stable representations, which are evaluated by the trapezoidal rule. In a second method an integral representation is evaluated by using generalized Gauss-Laguerre quadrature. This approach provides a fast method for computing Airy functions to a predetermined accuracy. Comparisons are made with well-known algorithms of Amos, designed for computing Bessel functions of complex argument. Several discrepancies with Amos’ code are detected, and it is pointed out for which regions of the complex plane Amos’ code is less accurate than the quadrature algorithms. Hints are given in order to build reliable software for complex Airy functions.

33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
33F05Numerical approximation and evaluation of special functions
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
30E10Approximation in the complex domain
65D20Computation of special functions, construction of tables
65D32Quadrature and cubature formulas (numerical methods)
Full Text: DOI