Eigenvalues for higher-order conical-horn modes

Macdonald´s formula, which is used to relate the associated Legendre and Bessel functions, is employed to find the approximate eigenvalues for TE and TM modes in a conical horn. Although the formula is restricted to small angles ¿, it nevertheless provides a very simple and accurate method for calculating the eigenvalues of the fundamental and higher-order modes, even for large values of ¿.