Conjugate function method for numerical conformal mappings

We present a method for numerical computation of conformal mappings from simply or doubly connected domains onto so-called canonical domains, which in our case are rectangles or annuli. The method is based on conjugate harmonic functions and properties of quadrilaterals. Several numerical examples are given.     

A linear operator associated with the Mittag-Leffer function and related conformal mappings

In the present paper, we introduce a linear operator associated with the Mittag-Leffler function. Some convolution properties of meromorphic functions involving this operator are given.     

Construction of conformal,mappings of biconnected regions

The author's method for conformal mapping of simply connected circular regions of a special form is extended to biconnected regions. Examples are given to show that the solution of the problem for some common biconnected regions may be reduced to examining the corresponding simply connected regions with boundaries along straight lines and circles. The solutions given in this paper are convenient for generating numerical results.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 71–75, 1986.     

Stability of weakly almost conformal mappings

We prove a stability of weakly almost conformal mappings in for not too far below the dimension by studying the -quasiconvex hull of the set of conformal matrices. The study is based on coercivity estimates from the nonlinear Hodge decompositions and reverse Hölder inequalities from the Ekeland variational principle.

     

On the conformal, concircular, and spin mappings of gravitational fields

A mapping ρ: of two Riemannian or pseudo-Riemannian spaces is called aspin mapping if for each geodesic curve γ in M_n its image ρ^oγ is a spin-curve in the space . In gravitational fields spin-curves describe the trajectories of uniformly accelerated particles of constant mass with simultaneous self-rotation. We prove: 1) a conformal mapping is a spin mapping only when it is concircular; 2) every conformal mapping of Einstein space is a spin mapping. The latter makes it possible to give a local representation of the metrics of all gravitational fields that admit spin mappings. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 44–47. Original article submitted March 17, 1993.     

Second derivatives of circle packings and conformal mappings

William Thurston conjectured that the Riemann mapping functionf from a simply connected region Ω onto the unit disk can be approximated as follows. Almost fill Ω with circles of radius ɛ packed in the regular hexagonal pattern. There is a combinatorially isomorphic packing of circles in . The correspondencef _ɛ of ɛ-circles in Ω with circles of varying radii in should converge tof after suitable normalization. This was proved in [RS], and in [H] an estimate was obtained which led to an approximation of |f′| in terms off _ɛ ; namely, |f′| is the limit of the ratio of the radii of a target circle off _ɛ to its source circle. In the present paper we show how to approximatef′ andf″ in terms off _ɛ . Explicit rates for the convergence tof, f′, andf″ are obtained. In the special case of convergence to |f′|, the estimate in this paper improves the previously known estimate.     

New extremal properties for constructing conformal mappings

Summary It is well known that for a given simply connected regionR containing zero the uniform norm attains its minimum in the class of all holomorphic functions normalized byf(0)=0 andf(0)=1 only for the conformal mappingfRD(r)={z|z|}. It is shown that this theorem is still valid if one replaces the ordinary modulus | | on by any other norm on . For instance it is possible to obtain direct mappings ofR onto parallelograms, rectangles and ellipses. For the special norms |₁ and | this leads to a simple and fast computational technique involving linear programming methods. Several numerical examples are given.