Non-Existence of Complex and Purely Imaginary Zeros of a Transcendental Equation Involving Bessel Functions

Abstract

Some results concerning the non-existence of complex and purely imaginary zeros of the transcendental equation $F(z)J_v(z)+G(z)J_v^{\prime }(z)=0$ are given, where $J_v(z)$ is the Bessel function of first kind and order $v$ (in general complex), $J_v^{\prime }(z)$ is the derivative of $J_v(z)$, and $F(z)$, $G(z)$ are analytic functions. The obtained results improve and generalize previously known ones.