Complex Methods for Three-Dimensional Magnetic Fields

A typical three-dimensional problem is to find the magnetic field off the end of a finite current multipole, i.e., a circular cylinder of finite length having a cos n¿ current distribution along its elements. The corresponding two-dimensional problem of finding the magnetic field both inside and outside of such a current 2n-pole of infinite length can be concisely solved by complex methods. This paper describes a way of systematically extending complex methods to three dimensions. The current multipole example is used to illustrate some of the features of the method. The results are given in closed form and involve complete elliptic integrals of all three kinds.