INTEGRAL FORMULAS ASSOCIATED WITH PRODUCTS OF BESSEL-FUNCTIONS - A NEW PARTIAL-DIFFERENTIAL EQUATION APPROACH

[摘要] We investigate the hyperbolic equation with two singular lines GRAPHICS for \alpha\ < 1/2, under the assumption that a certain asymptotic behavior of u and its partial derivates is prescribed for (xi, eta) tending to the coordinate axes or to infinity. Intending to establish integral formulas for products of Bessel functions which a priori solve such a problem, we are looking for an unusual representation of the solution u(x, y), 0 < x, y < infinity, in terms of ''initial data'' outside the interval \x - y\ less-than-or-equal-to xi less-than-or-equal-to x + y. This will be achieved essentially by choosing a novel geometric configuration of six subregions of the domain, to which Green's theorem will be applied. Moreover, we employ detailed knowledge about the corresponding Riemann function and a further auxiliary function associated to it. In this way, various product formulas are obtained, one of which is equivalent to a recent result of Askey et al. (1986), who evaluated an integral over a triple product of Bessel functions of the first and second kind. We also give a new proof of a product formula for modified Bessel functions due to Durand.

[发布日期] 1994-06-20

[关键词] BESSEL FUNCTIONS OF THE 1ST AND 2ND KIND;PARTIAL DIFFERENTIAL EQUATION;PRODUCT FORMULA;RIEMANN METHOD;ASSOCIATED RIEMANN FUNCTION