Distribution properties of the fringing field from a semi-infinite ring head with an arbitrary pole-angle

General formulae for studying the distribution of the fringing field from a semi-infinite ring head with an arbitrary pole-angle, are derived by the method of the Schwartz - Christoffel transformation. The mapping functions can be calculated without any approximation. Based on these formulae, the distributive properties of the fringing field are studied in detail. It is found that the field intensity is monotonic with respect to the pole angle, while the optimum one does not exist. It is also found that the Karlquist's formula, which was obtained on the assumption that the magnetic scalar potential has a constant gradient across the top of the gap, can be applicable at the practical recording distance only for a rectangular head.