The fractional linear transformations provide mappings between analytic functions satisfying the homogeneity and Herglotz properties. The field equation recursion method provides mappings at a deeper level, namely, at the level of the underlying Hilbert space. The field equation recursion method has two advantages over the analytic method: First, it has a natural generalization to multiphase composites, and, second, the method provides matrix representations for the relevant operators. Also, with the introduction of additional fields and operators, the method allows one to incorporate the differential constraints on the fields in a direct fashion.

The original papers (Milton 1987a, 1987b) describing the field equation recursion method are difficult to read, but fortunately many of the arguments have since been simplified. It is hoped that the presentation given here will convey the main ideas; see also Milton (1991), where a brief summary is given. We will focus on two-phase composites, since the analysis is simpler and one can immediately see the connection with the analytic method. Another simple case that we could have discussed is the conductivity of a two-dimensional polycrystal, where Clark (1997) has shown that the field equation recursion method leads to an elegant continued fraction expansion of the effective conductivity tensor as a function of the crystal conductivity.