## Abstract

Let C_{t} = {z ∈ ℂ: {pipe}z - c(t){pipe} = r(t), t ∈ (0, 1)} be a C^{1}-family of circles in the plane such that lim_{t→0+}C_{t} = {a}, lim_{t→1-}C_{t} = {b}, a ≠ b, and {pipe}c′(t){pipe}^{2} + {pipe}r′(t){pipe}^{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 c̄′(t)w^{2} + 2r′(t)w + c′(t) = 0 with {pipe}w{pipe} < 1, if such a root exists. Suppose that S does not contain a continuous curve joining a and b. We prove that if a smooth and regular (in a certain sense) function f in Ω = ∪C_{t}, possesses, for each t ∈ (0, 1), a meromorphic extension inside C_{t} whose only singular point is a pole at c(t) of order at most ν ∈ {0} ∪ ℕ, then f is polyanalytic of order at most ν, i. e., f(z) = h_{0}(z) + h_{1}(z)z̄+̇̇̇z̄^{ν}h_{ν}(z), where the h′_{j}s are analytic functions in Ω. For ν = 0, the condition on S can be omitted. If {pipe}c′(t){pipe} > {pipe}r′(t){pipe}, t ∈ (0, 1), then S = ∅ and the condition of regularity can be dropped. A hyperbolic version of the result is also given.

Original language | English |
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Pages (from-to) | 305-329 |

Number of pages | 25 |

Journal | Journal d'Analyse Mathematique |

Volume | 113 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2011 |

### Bibliographical note

Funding Information:∗This work was partially supported by ISF (Israel Science Foundation), Grant 688/08. Some of this research was done as a part of European Networking Program HCAA.