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

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