ZEROS' DISTRIBUTION OF THE FIRST KIND BESSEL FUNCTIONS

The aim of this paper is to investigate the zeros' distribution of the first kind Bessel functions J(v)(z) of order v >= 1. The problem arises from the conjecture given by the work [8] which considered the existence of smooth solutions for one-dimensional compressible Euler equation with gravity. In this article we show that J(v)(L theta.) not equal 0 for any integer L >= 2 provided that J(v)(theta) = 0, v >= 1 and theta is sufficiently large. Moreover, if. is half of an odd integer, we can remove the restriction of large theta and show that J(v)(L theta.) not equal 0 for any integer L >= 2.