Two new variants of the -transformation due to the author [J. Inst. Math. Appl. 26, 1-20 (1980; Zbl 0464.65002)] are designed for computing oscillatory integrals , where is a non-oscillatory function and stands for an arbitrary linear combination of the Bessel functions of the first and second kinds and , of arbitrary real order .
The author also points out that the application to such integrals of an additional approach involving the so-called -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 -transformation and the -transformation in all of their forms are discussed and a numerical example is added.