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Applications of some recent results in asymptotic expansions. (English) Zbl 0539.41030
Numerical mathematics and computing, Proc. 12th Manitoba Conf., Winnipeg/Manit. 1982, Congr. Numerantium 37, 145-182 (1983).
[For the entire collection see Zbl 0532.00009.] In this paper, infinite asymptotic expansions are derived for the following three integrals: (i) $\int\sp{\infty}\sb{0}e\sp{-\lambda t\sp 2}(\sin t/t\sp 2)J\sb 1(t)dt,$ where $J\sb 1(t)$ is the Bessel function of the first kind of order 1 and $\lambda$ is a positive parameter tending to zero; (ii) $\int\sp{\infty}\sb{0}dt/\sqrt{(t+x)(t+y)(t+z)},$ where x and y are fixed positive numbers and z is a positive parameter tending to infinity; (iii) $\int\sp{1}\sb{0}(t/\sqrt{1-t\sp 2})e\sp{- 2ixt\sp 2}F(2\sqrt{x/\pi}t)dt,$ where F(t) is expressible in terms of the complementary error function of argument $e\sp{-i\pi /4}t$ and x is a large positive parameter. The method for the first integral is based on results from the Mellin transform theory, and the methods for the second and third integrals make use of the theory of distributions.
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)