The asymptotic behavior of conformal modules of quadrilaterals with applications to the estimation of resistance values
If a quadrilateral \(Q\) is subdivided by crosscuts in \(Q\) into component quadrilaterals \(Q_j\) and \(m(Q)\), \(m(Q_j)\) are their conformal modules, the composition law gives \(m(Q)\geq \sum_j m(Q_j)\). One aspect of the present paper is to present asymptotic error estimates for \(m(Q)-\sum_jm(Q_j)\) in case that \(Q\) is `long' in one or two directions. These error estimates use previous estimates by Gaier and Hayman (1991) for simpler geometries. A special feature is the comparison of the authors' results with engineering rules that are used in the construction of integrated circuit networks. This is possible because \(m(Q)\) can be interpreted as the resistance of a conducting plate. These rules can now be used with error estimates. In a numerical example, \(Q\) is a polygonal domain with 30 corners, and \(m(Q)\) is determined with six correct decimals.
scientific article; zbMATH DE number 1271886