The singularities of a Fourier-type integral in a multicylindrical layer problem

The singularities of the integrand of a Fourier-type integral obtained in solving the multicylindrical layer boundary value problem are discussed. The integrand is a function of the radial wavenumber kᵢₚ of all the cylindrical layers, and the radial wavenumber in the ith layer is related to the axial wavenumber bykᵢₚ = √{kᵢ² - k²}wherekᵢis the wavenumber of theith layer, andkzis the axial wavenumber of all the layers which have to be the same by phase matching. On the complexkz-plane, there seemingly are branch points of logarithmic type and algebraic type forkz = kᵢfor all the layers. However, by invoking uniqueness principle in the solution of this boundary value problem, one can show that the only singularities on the complexkz- plane are the branch-point singularity associated with the outermost medium which extends radially to infinity, and pole singularities which correspond to discrete guided modes in the multicylindrical medium.