# 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)
Computation of infinite integrals involving Bessel functions of arbitrary order by the $\overline{D}$-transformation. (English) Zbl 0871.65012
Two new variants of the $\overline D$-transformation due to the author [J. Inst. Math. Appl. 26, 1-20 (1980; Zbl 0464.65002)] are designed for computing oscillatory integrals $\int^\infty_ag(t)\cdot{\cal C}_\nu(t)dt$, where $g(t)$ is a non-oscillatory function and ${\cal C}_\nu (x)$ stands for an arbitrary linear combination of the Bessel functions of the first and second kinds $J_\nu(x)$ and $Y_\nu(x)$, of arbitrary real order $\nu$. The author also points out that the application to such integrals of an additional approach involving the so-called $mW$-transformation as well introduced by himself [see Math. Comput. 51, No. 183, 249-266 (1988; Zbl 0694.40004); see also {\it D. Levin} and {\it A. Sidi}, Appl. Math. Comput. 9, 175-215 (1981; Zbl 0487.65003)] produces similarly good results. Finally, convergence and stability results of both the $\overline D$-transformation and the $mW$-transformation in all of their forms are discussed and a numerical example is added.

##### MSC:
 65D20 Computation of special functions, construction of tables 33C10 Bessel and Airy functions, cylinder functions, ${}_0F_1$ 33E20 Functions defined by series and integrals 65D32 Quadrature and cubature formulas (numerical methods)
Full Text:
##### References:
 [1] Abramowitz, M.; Stegun, I. A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. Nat. bur. Standards appl. Math. series 55 (1964) · Zbl 0171.38503 [2] Ehrenmark, U. T.: The numerical inversion of two classes of kontorovich-lebedev transform by direct quadrature. J. comput. Appl. math. 61, 43-72 (1995) · Zbl 0841.65124 [3] Gradshteyn, I. S.; Ryzhik, I. M.: Table of integrals, series and products. (1983) · Zbl 0918.65002 [4] Hasegawa, T.; Sidi, A.: An automatic integration procedure for infinite range integrals involving oscillatory kernels. Numer. algorithms 13, 1-19 (1996) · Zbl 0879.65015 [5] Hasegawa, T.; Torii, T.: Indefinite integration of oscillatory functions by the Chebyshev series expansion. J. comput. Appl. math. 17, 21-29 (1987) · Zbl 0613.65145 [6] Levin, D.; Sidi, A.: Two new classes of nonlinear transformations for accelerating the convergence of infinite integrals and series. Appl. math. Comput. 9, 175-215 (1981) · Zbl 0487.65003 [7] Lucas, S. K.; Stone, H. A.: Evaluating infinite integrals involving Bessel functions of arbitrary order. J. comput. Appl. math. 64, 217-231 (1995) · Zbl 0857.65025 [8] Lyness, J. N.: Integrating some infinite oscillating tails. J. comput. Appl. math. 12 and 13, 109-117 (1985) · Zbl 0574.65013 [9] Sidi, A.: Some properties of a generalization of the Richardson extrapolation process. J. inst. Math. appl. 24, 327-346 (1979) · Zbl 0449.65001 [10] Sidi, A.: Extrapolation methods for oscillatory infinite integrals. J. inst. Math. appl. 26, 1-20 (1980) · Zbl 0464.65002 [11] Sidi, A.: The numerical evaluation of very oscillatory infinite integrals by extrapolation. Math. comput. 38, 517-529 (1982) · Zbl 0508.65011 [12] Sidi, A.: An algorithm for a special case of a generalization of the Richardson extrapolation process. Numer. math. 38, 299-307 (1982) · Zbl 0485.65004 [13] Sidi, A.: Extrapolation methods for divergent oscillatory infinite integrals that are defined in the sense of summability. J. comput. Appl. math. 17, 105-114 (1987) · Zbl 0634.41018 [14] Sidi, A.: A user-friendly extrapolation method for oscillatory infinite integrals. Math. comput. 51, 249-266 (1988) · Zbl 0694.40004 [15] Sidi, A.: On rates of acceleration of extrapolation methods for oscillatory infinite integrals. Bit 30, 347-357 (1990) · Zbl 0714.65021