Transverse problem in coaxial regions leads to the well known Bessel equation. Solution of this equation is in the form of Fourier-Bessel expansion of which eigenvalues are determined by the appropriate transcendental equations. In electromagnetics, two symmetric cases: Dirchlet-Dirichlet (D-D) and Neumann-Neumann (N-N) are fundamental both from practical and theoretical points of view. The first gives rise to TMmn modes and the second gives rise to TEmn modes. The first boundary value problem lead to the following transcendental equation for the eigenvalues of the Fourier-Bessel expansion jm(χmn)Ym(χm nℓ)-Jm(χm nℓ)Ym(χm n) = 0 (1) and the second boundary value problem leads to J'm (χm n) Y'm(χm nℓ) − J'm(χm nℓ) Y'm(χmn) = 0 (2) where the prime' denotes derivative with respect to the argument.