Roots of $J_\gamma (z)\pm iJ_{\gamma +1}(z)=0$ and the evaluation of integrals with cylindrical function kernels

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