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 k_{ip} of all the cylindrical layers, and the radial wavenumber in the ith layer is related to the axial wavenumber by where is the wavenumber of the th layer, and is the axial wavenumber of all the layers which have to be the same by phase matching. On the complex -plane, there seemingly are branch points of logarithmic type and algebraic type for for 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 complex - 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.