Equivalent line current for cylindrical dipole antennas and its asymptotic behavior

The concept of an equivalent line source for representing the current on a cylindrical dipole antenna is introduced. It is shown that the Fourier series coefficients of this equivalent line source are related to those of the actual antenna current by exponentially growing factors that grow rapidly for the higher order harmonics. This is used to explain why Hallen's integral equation using an approximate kernel does not have an exact solution. The asymptotic behavior of the Fourier coefficients is established. An explanation of why approximate solutions to the approximate integral equation often provide good results for the current and input impedance is also given.