The Schwarz Function and Its Applications

A four-dimensional Neumann ovaloid

“…For further details see [2] and [6]. An equivalent definition of the quadrature domain (1) is by the free boundary problem (5) Δu=T, in Ω u= 1 2 |x| 2 , ∇u=x on ∂Ω .…”

“…In particular, when T is a measure, then by applying (1) to the Newtonian kernel it results that the external potential of the body Ω with density one is equal to the potential of the measure T . If T is a finitely-supported distribution of finite-order (so the right-hand-side of (1) is a finite sum of weighted point evaluations of u and its partial derivatives), then Ω is referred to as a quadrature domain in the classical sense.…”

Cauchy, Goursat and Dirichlet Problems for Holomorphic Partial Differential Equations

“…In [29] the function S is called the Schwarz function and the reader will find there a detailed account, many examples and applications to various areas in complex function theory.…”

“…Section 6 is devoted to a short introduction to the Schwarz potential in R d which generalizes the Schwarz function known from the two-dimensional case, see [29]. The interested reader is referred to the excellent expositions [64], [65], [69] and [91] for a deeper analysis.…”

О Зависимости Условий Равномерной Приближаемости Функций Полианалитическими Многочленами От Порядка Полианалитичности

“…Напомним, что для любого аналитиче-ского контура Γ существует функция Шварца (см. [6]), т.е. такая голоморфная и однолистная в некоторой области U ⊃ Γ функция S, что Γ = {z ∈ U : z = S(z)}.…”