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Computation of infinite integrals involving Bessel functions of arbitrary order by the D ¯-transformation. (English) Zbl 0871.65012

Two new variants of the D ¯-transformation due to the author [J. Inst. Math. Appl. 26, 1-20 (1980; Zbl 0464.65002)] are designed for computing oscillatory integrals a g(t)·𝒞 ν (t)dt, where g(t) is a non-oscillatory function and 𝒞 ν (x) stands for an arbitrary linear combination of the Bessel functions of the first and second kinds J ν (x) and Y ν (x), of arbitrary real order ν.

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 D. Levin and A. Sidi, Appl. Math. Comput. 9, 175-215 (1981; Zbl 0487.65003)] produces similarly good results. Finally, convergence and stability results of both the D ¯-transformation and the mW-transformation in all of their forms are discussed and a numerical example is added.

65D20Computation of special functions, construction of tables
33C10Bessel and Airy functions, cylinder functions, 0 F 1
33E20Functions defined by series and integrals
65D32Quadrature and cubature formulas (numerical methods)