Roots of 𝐽ᵧ(𝑧)±𝑖𝐽ᵧ₊₁(𝑧)=0 and the evaluation of integrals with cylindrical function kernels

Abstract

An elementary proof is presented showing that the function f ( z ) = J γ ( z ) ± i J γ + 1 ( z ) f\left ( z \right ) = \\ {J_\gamma }\left ( z \right ) \pm i{J_{\gamma + 1}} \left ( z \right ) , where γ \gamma is a natural number, has no zeroes in the lower and upper half-planes respectively. The roots of f ( z ) f\left ( z \right ) are given for certain values of γ \gamma and their locations are plotted. Cartesian maps (mappings of constant coordinate lines) of f ( z ) f\left ( z \right ) are obtained, and special features of these maps are discussed. Some integrals with cylindrical kernels involving f ( z ) f\left ( z \right ) are obtained in terms of the zeroes of J γ ( z ) {J_\gamma }\left ( z \right ) .