A magnetic double integral

TL;DR

This paper explores a complex double integral related to Hall plate voltage, examining its transformation properties, analytical features, and connections to modular forms, extending classical elliptic integrals.

Contribution

It introduces a generalized double integral linked to Hall plate analysis and investigates its mathematical properties and connections to modular forms.

Findings

Transformation properties observed experimentally

Analytical features characterized mathematically

Connections established with modular forms

Abstract

In a recent study of how the output voltage of a Hall plate is affected by the shape of the plate and the size of its contacts, Udo Ausserlechner has come up with a remarkable double integral that can be viewed as a generalization of the classical elliptic "AGM" integral. Here we discuss transformation properties of the integral, which were experimentally observed by Ausserlechner, as well as its analytical and arithmetic features including connections with modular forms.