Nicolas Papamichael
Title :
The Computation of Conformal Modules by Domain Decomposition
Abstract :
The conformal module m(Q) of a quadrilateral Q := {Ω; z1, z2, z3, z4}, consisting of a Jordan
domain Ω and four points z1, z2, z3, z4, in counter-clockwise order on ∂Ω , is defined as
follows:
Let Rh denote a rectangle of length 1 and height h of the form Rh := {w : 0 < Re w < 1; 0 < Im
w < h }. Then m(Q) is the unique value of h for which the quadrilateral Q is conformally
equivalent to the rectangular quadrilateral {Rh ; 0, 1, 1 + ih, ih}, in the sense that for h =
m(Q) and for this value only there exists a unique conformal mapping F : Ω  Rh that takes
the four points z1, z2, z3, z4 respectively onto the four vertices 0, 1, 1 + ih, ih of Rh.
It turns out that the conformal module m(Q) is a theoretically important domain functional
that plays a significant role in geometric function theory. At the same time, the module is
closely related to certain physical constants that occur in engineering applications, notably
in connection with the measurement of resistance values of integrated circuit networks.
In this talk we describe a simple procedure for computing (essentially by hand calculation)
accurate approximations to the conformal modules of complicated quadrilaterals of the
type that occur frequently in engineering applications. (For example, in VLSI applications
involving the measurement of resistance values of integrated circuit networks.) The
procedure is based on the use of certain asymptotic formulas from the engineering
literature (that date back to 1967) in conjunction with more recent results from the
theories of: (i) a domain decomposition method for computing the modules of long
quadrilaterals, and(ii) the asymptotic behavior of the modules of strip-like quadrilaterals.
(This is a report of joint work with N.S. Stylianopoulos.)