The Schottky-Klein Prime Function on the Schottky Double of Planar Domains

2010 ◽  
Vol 10 (2) ◽  
pp. 501-517 ◽  
Author(s):  
Darren Crowdy
Keyword(s):  
Planar Domains ◽  
Schottky Double ◽  
Prime Function

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

2015 ◽  
Vol 471 (2173) ◽  
pp. 20140688 ◽  
Author(s):  
Giovani L. Vasconcelos ◽  
Jonathan S. Marshall ◽  
Darren G. Crowdy
Keyword(s):  
Mixed Type ◽  
Compact Riemann Surface ◽  
Mathematical Framework ◽  
Multiply Connected Domains ◽  
Additional Function ◽  
Multiply Connected ◽  
Planar Domains ◽  
Recent Interest ◽  
Schottky Double ◽  
Prime Function

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.

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

2021 ◽  
Author(s):  
Hiroyuki Miyoshi ◽  
Darren Crowdy ◽  
Rhodri Nelson
Keyword(s):  
Applied Sciences ◽  
Science Literature ◽  
Multiply Connected ◽  
Schottky Double ◽  
Van Der Pauw ◽  
Simply Connected ◽  
Genus One ◽  
Physical Quantities ◽  
Prime Function ◽  
Van Der Pauw Method

AbstractThe 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.

scholarly journals Stress fields around two pores in an elastic body: exact quadrature domain solutions

2015 ◽  
Vol 471 (2180) ◽  
pp. 20150240
Author(s):  
Darren Crowdy
Keyword(s):  
Elastic Body ◽  
Analytical Solutions ◽  
Plane Elasticity ◽  
Stress Fields ◽  
Quadrature Domain ◽  
Mathematical Approach ◽  
Quadrature Domains ◽  
Planar Domains ◽  
Elastic Bodies ◽  
Prime Function

Analytical solutions are given for the stress fields, in both compression and far-field shear, in a two-dimensional elastic body containing two interacting non-circular pores. The two complex potentials governing the solutions are found by using a conformal mapping from a pre-image annulus with those potentials expressed in terms of the Schottky–Klein prime function for the annulus. Solutions for a three-parameter family of elastic bodies with two equal symmetric pores are presented and the compressibility of a special family of pore pairs is studied in detail. The methodology extends to two unequal pores. The importance for boundary value problems of plane elasticity of a special class of planar domains known as quadrature domains is also elucidated. This observation provides the route to generalization of the mathematical approach here to finding analytical solutions for the stress fields in bodies containing any finite number of pores.

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

2021 ◽  
Vol 477 (2246) ◽  
pp. 20200796
Author(s):  
Hiroyuki Miyoshi ◽  
Darren G. Crowdy ◽  
Rhodri Nelson
Keyword(s):  
Applied Sciences ◽  
Outer Boundary ◽  
Recent Interest ◽  
Schottky Double ◽  
Explicit Formulae ◽  
Van Der Pauw ◽  
Simply Connected ◽  
Prime Function ◽  
Made In ◽  
Van Der Pauw Method

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.

scholarly journals Euler Equations on General Planar Domains

Annals of PDE ◽  
2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Zonglin Han ◽  
Andrej Zlatoš
Keyword(s):  
Euler Equations ◽  
Planar Domains

scholarly journals Polyanalytic boundary value problems for planar domains with harmonic Green function

2021 ◽  
Vol 11 (3) ◽  
Author(s):  
Heinrich Begehr ◽  
Bibinur Shupeyeva
Keyword(s):  
Green Function ◽  
Boundary Value Problems ◽  
Arbitrary Order ◽  
Boundary Value ◽  
Planar Domains ◽  
Schwarz Problem ◽  
Bianalytic Functions ◽  
Neumann Type ◽  
The Neumann Problem ◽  
Well Posed

AbstractThere are three basic boundary value problems for the inhomogeneous polyanalytic equation in planar domains, the well-posed iterated Schwarz problem, and further two over-determined iterated problems of Dirichlet and Neumann type. These problems are investigated in planar domains having a harmonic Green function. For the Schwarz problem, treated earlier [Ü. Aksoy, H. Begehr, A.O. Çelebi, AV Bitsadze’s observation on bianalytic functions and the Schwarz problem. Complex Var Elliptic Equ 64(8): 1257–1274 (2019)], just a modification is mentioned. While the Dirichlet problem is completely discussed for arbitrary order, the Neumann problem is just handled for order up to three. But a generalization to arbitrary order is likely.

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