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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.

65D20Computation of special functions, construction of tables
33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
33E20Functions defined by series and integrals
65D32Quadrature and cubature formulas (numerical methods)
Full Text: DOI
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