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.

Secondary Schottky–Klein prime functions associated with multiply connected planar domains

In recent years, a general mathematical framework for solving applied problems in multiply connected domains has been developed based on use of the Schottky–Klein (S–K) prime function of an underlying compact Riemann surface known as the Schottky double of the domain. In this paper, we describe additional function-theoretic objects that are naturally associated with planar multiply connected domains and which we refer to as secondary S–K prime functions. The basic idea develops, and extends, an observation of Burnside dating back to 1892. Applications of the new functions to represent conformal slit maps of mixed type that have been a topic of recent interest in the literature are given. Other possible applications are also surveyed.

BOUNDARY REPARAMETRIZATIONS AS ADDITIONAL MODULI FOR THE STRING PROPAGATOR

Analyzing the Polyakov integral on surfaces with boundaries, where the values of the string variables are fixed, we use the observation that there are more holomorphic quadratic differentials besides those obtained as restrictions from the Schottky double. They are naturally related to boundary reparametrizations. The corresponding additional moduli are used to express the integration over metrices. Some details are given for the vacuum functional and the propagator.