Fay meets van der Pauw: the trisecant identity and the resistivity of holey samples

The van der Pauw method is commonly used in the applied sciences to find resistivity of a simply connected, two-dimensional conducting laminate. Given usefulness this ‘4-point probe’ there has been much recent interest trying extend it holey, that is, multiply samples. This paper introduces two new mathematical tools area investigation—the prime function on Schottky double planar domain and Fay trisecant identity—and uses them show how can be extended sample with hole. We an integrated form identity provides valuable information concerning appearance ‘envelopes’ observed case holey samples by previous authors. explicit formulae for these envelopes, as well approximate formula relating pairs resistance measurements expected valid when hole sufficiently small not too close outer boundary. describe have enabled us prove certain conjectures recently made engineering literature.