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Zbl 0579.41028
Frenzen, C.L.; Wong, R.
A note on asymptotic evaluation of some Hankel transforms. (English)
[J] Math. Comput. 45, 537-548 (1985). ISSN 0025-5718; ISSN 1088-6842/e

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.

MSC 2000:: *41A60 Asymptotic problems in approximation
44A15 Special transforms

Keywords: Hankel transform; Laplace's method; Bessel function; meromorphic function; growth condition; branch point singularities

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