The Prime Function, the Fay Trisecant Identity, and the van der Pauw Method

The van der Pauw method is a well-known experimental technique in the applied sciences for measuring physical quantities such as the electrical conductivity or the Hall coefficient of a given sample. Its popularity is attributable to its flexibility: the same method works for planar samples of any shape provided they are simply connected. Mathematically, the method is based on the cross-ratio identity. Much recent work has been done by applied scientists attempting to extend the van der Pauw method to samples with holes (“holey samples”). In this article we show the relevance of two new function theoretic ingredients to this area of application: the prime function associated with the Schottky double of a multiply connected planar domain and the Fay trisecant identity involving that prime function. We focus here on the single-hole (doubly connected, or genus one) case. Using these new theoretical ingredients we are able to prove several mathematical conjectures put forward in the applied science literature.

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

The anisotropic structure of electro conductive leather studied by Van der Pauw method

Determining the surface resistance of electro conductive refined natural leather materials is in the focus of this paper. Natural leather samples are initially transformed to conductive by applying chemical treatment process known as polymerization. Due to the existence of various techniques for measuring electrical resistance of conductive materials, we are focused on measuring surface resistance by arranging four electrodes in the edges of square leather samples, also known as Van der Pauw method. Improving the results accuracy, we use a multi-variant electrode placement over the sample edges. The result is the average of all results gained for different placements. Moreover, we use this electrode placement technique to analyse the anisotropy of conductive samples. The results of this research provide important knowledge about leather chemical treatment and its electrical proprieties.