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A note on asymptotic evaluation of some Hankel transforms. (English) Zbl 0579.41028
Summary: Asymptotic behavior of the integral $$ I\sb f(w)=\int\sp{\infty}\sb{0}e\sp{-x\sp 2}J\sb 0(wx)f(x\sp 2)x dx $$ is investigated, where $J\sb 0(x)$ is the Bessel function of the first kind and w is a large positive parameter. It is shown that $I\sb f(w)$ decays exponentially like $e\sp{-\gamma w\sp 2}$, $\gamma >0$, when f(z) is an entire function subject to a suitable growth condition. A complete asymptotic expansion is obtained when f(z) is a meromorphic function satisfying the same growth condition. Similar results are given when f(z) has some specific branch point singularities.

41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
44A15Special transforms (Legendre, Hilbert, etc.)
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