# 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)
Modified Clenshaw-Curtis method for the computation of Bessel function integrals. (English) Zbl 0514.65008
##### MSC:
 65D20 Computation of special functions, construction of tables 65T40 Trigonometric approximation and interpolation (numerical methods) 65D32 Quadrature and cubature formulas (numerical methods) 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 41A55 Approximate quadratures 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 65R10 Integral transforms (numerical methods)
##### References:
 [1] M. Abramowitz and I. Stegun (Eds.),Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover Publ., New York (1965), p. 364. [2] W. L. Anderson,Fast Hankel transforms using related and lagged convolutions, TOMS 8 (1982), 344–368. · Zbl 0493.65071 · doi:10.1145/356012.356014 [3] M. Branders,The asymptotic behaviour of solutions of difference equations, Bull. Soc. Math. Belg. 26 (1974), 255–260. [4] M. Branders and R. Piessens,Algorithm 001, An extension of Clenshaw-Curtis quadrature, J. CAM 1 (1975), 55–65. [5] C. W. Clenshaw,Curve fitting with a digital computer, Computer J. 2 (1960), 170–173. · Zbl 0095.11801 · doi:10.1093/comjnl/2.4.170 [6] C. W. Clenshaw and A. R. Curtis,A method for numerical integration on an automatic computer, Num. Math. 2 (1960), 197–205. · Zbl 0093.14006 · doi:10.1007/BF01386223 [7] P. Cornille,Computation of Hankel transforms, SIAM Rev. 14 (1972), 278–285. · Zbl 0234.65117 · doi:10.1137/1014032 [8] B. Davies,Integral Transforms and their Applications, Springer-Verlag, Heidelberg (1978), pp. 237–247. [9] P. J. Davis and P. Rabinowitz,Methods of Numerical Integration, Academic Press, New York (1975), pp. 118–132 and pp. 178–180. [10] J. Denef and R. Piessens,The asymptotic behaviour of solutions of difference equations of Poincaré:s type, Bull. Soc. Math. Belg. 26 (1974), 133–146. [11] A. Erdélyi,Asymptotic representations of Fourier integrals and the method of stationary phase, J. Soc. Industr. Appl. Math. 3 (1955), 17–27. · Zbl 0072.11702 · doi:10.1137/0103002 [12] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi,Tables of Integral Transforms, vol. 2, McGraw-Hill, New York (1954), p. 5. [13] W. Gautschi,Algorithm 236: Bessel functions of the first kind, Comm. ACM 7 (1964), 479–480. · doi:10.1145/355586.355587 [14] W. Gautschi,Computational aspects of three-term recurrence relations, SIAM Rev. 9 (1967), 24–82. · Zbl 0168.15004 · doi:10.1137/1009002 [15] W. Gautschi,On the construction of Gaussian rules from modified moments, Math. Comp. 24 (1970), 245–260. [16] W. M. Gentleman,Algorithm 424, Clenshaw-Curtis quadrature, Comm. ACM 15 (1972), 353–355. · doi:10.1145/355602.355603 [17] P. Linz,A method for computing Bessel function integrals, Math. Comp. 26 (1972), 504–513. · doi:10.1090/S0025-5718-1972-0303687-8 [18] I. M. Longman,A short table of x J 0(t)t dt and x J 1(t)t dt, MTAC 13 (1959), 306–311. [19] I. M. Longman,A method for the numerical evaluation of finite integrals of oscillatory functions, Math. Comp. 14 (1960), 53–59. · doi:10.1090/S0025-5718-1960-0111136-X [20] Y. L. Luke,The special functions and their approximations, vol. 2, Academic Press, New York and London (1969), p. 53 and p. 334. [21] J. Oliver,The numerical solution of linear recurrence relations, Num. Math. 11 (1968), 349–360. · Zbl 0164.45401 · doi:10.1007/BF02166688 [22] R. Piessens,Gaussian quadrature formulas for integrals involving Bessel functions, Microfiche section of Math. Comp. 26 (1972). [23] R. Piessens and M. Branders,Approximations for Bessel functions and their application in the computation of Hankel transforms, Comp. & Maths. with Appls. 8 (1982), 305–311. · Zbl 0482.65068 · doi:10.1016/0898-1221(82)90012-8 [24] R. Piessens, E. de Doncker, C. Überhuber and D. Kahaner,QUADPACK: A Subroutine Package for Numerical Integration, to be published by Springer-Verlag, Heidelberg, pp. 66–67. [25] M. A. Snyder,Chebyshev Methods in Numerical Approximation, Prentice-Hall, Englewood Cliffs, N.J. (1966), p. 103. [26] G. N. Watson,Treatise on the Theory of Bessel Functions, Cambridge University Press, 2nd ed. (1958), pp. 537–538.