Asymptotic expansion of a quadruple integral involving a Bessel function
The quadruple integral \[ Q(\lambda):=\int^{\pi /2}_{0}...\int^{\pi /2}_{0}I_ 0(\lambda [\cos \alpha_ 1\quad \cos \alpha_ 2+\cos \alpha_ 3\quad \cos \alpha_ 4])d\alpha_ 1d\alpha_ 2d\alpha_ 3d\alpha_ 4, \] which occurs in crystallographic problems, is investigated with respect to its asymptotic behavior for \(\lambda\to \infty\), \(\lambda\in {\mathbb{R}}\). Here \(I_ 0\) denotes the ordinary Bessel function of order zero. The authors develop a new method of constructing an asymptotic expansion, where the first step reduces the integral to a finite Hankel transform. The second step consists of deriving an asymptotic expansion for this finite transform. The details of this approach are quite complicated. A lot of special results from classical analysis, e.g. analytic properties of some Mellin transforms and of the Gamma function, analytic continuation techniques for some meromorphic functions, are necessary for proving the asymptotic results. Some more interesting examples are included in the paper in order to show the wide applicability of the new method.
scientific article; zbMATH DE number 4192347
scientific article