Although the King-Middleton second-order solution for the input impedance of a center-fed cylindrical antenna cannot be considered completely "correct", it can be taken as a reference to validate a simulation method.
The King-Middleton analytical solution is based on an expansion of the current distribution along a linear antenna in terms of sinusoidal functions (and integrals of those functions) and neglects edge effects at the feed point (such as the finiteness of the gap). Needless to say, it is impressive that an analytical second-order iteration of the solution of an integral equation could be obtained in King's days.
The figure shows King-Middleton results compared to those obtained from the modern implementation of the Method of Moments (MoM) in AN-SOF. Rin (Xin) is the real (imaginary) part of the input impedance. Beta is the wavenumber, h is the dipole half-length. For a wavelength of 1 m, the antenna diameter is (a) 3/32 in, (b) 1/4 in, (c) 3/4 in.
As can be seen a very good agreement is achieved between the King-Middleton second-order approximation and the results obtained from AN-SOF. These results will be published in the validation manual that we are writing.