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, U. Ausserlechner has come up with a remarkable double integral that can be viewed as a generalisation of the classical elliptic ‘arithmetic–geometric mean (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.

Highlights

  • Recall that the arithmetic–geometric mean (AGM) agm(a, b) of two positive real numbers a and b is defined as a common limit of the sequences a0 = a, a1, a2, . . . and b0 = b, b1, b2, . . . generated by the iteration an+1

  • The AGM sources several beautiful structures, formulae and algorithms in mathematics; it is linked with periods of elliptic curves, modular forms and the hypergeometric function

  • 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, Ausserlechner [1] has come up with the double integral

Read more

Summary

Introduction

Recall that the arithmetic–geometric mean (AGM) agm(a, b) of two positive real numbers a and b is defined as a common limit of the sequences a0 = a, a1, a2, . Dα cos α + f 2 sin α c 2019 Australian Mathematical Publishing Association Inc. dα cos α + f 2 sin α c 2019 Australian Mathematical Publishing Association Inc Is a particular instance of a classical elliptic integral. The AGM sources several beautiful structures, formulae and algorithms in mathematics; it is linked with periods of elliptic curves, modular forms and the hypergeometric function. Though the method in [6] is seemingly less laborious than the strategy below, our approach is different and reveals further structure of I2( f )

Preliminaries
Numerical evaluation of the double integral
The inhomogeneous differential equation
Findings
Final remarks

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.