Characterization of polyanalytic functions by meromorphic extensions from chains of circles

Abstract

Let Ct = {z ∈ ℂ: |z − c(t)| = r(t), t ∈ (0, 1)} be a C1-family of circles in the plane such that limt→0+Ct = {a}, limt→1−Ct = {b}, a ≠ b, and |c′(t)|2 + |r′(t)|2 ≠ 0. The discriminant set S of the family is defined as the closure of the set {c(t) + r(t)w(t), t ∈ [0, 1]}, where w = w(t) is the root of the quadratic equation c′(t)w2 + 2r′(t)w + c′(t) = 0 with |w| < 1, if such a root exists.