Picard Sets

Abstract

For the set of Picard values of a meromorphic function at a point z₀, i.e., values omitted by it in a neighborhood of z ₀, we shall use in this book the suggestive term Picard set. We know (IV.7B) that the Picard set of a meromorphic function at each point of a set of essential singularities of vanishing capacity is again a set with this property. We shall show in §1 that this result is the best possible: for an arbitrary compact set K of capacity zero there exists a meromorphic function f with exactly K as its Picard set at every essential singularity, the set E of these singularities having vanishing capacity. On the other hand, if E has positive capacity, we can construct an f such that its Picard set at each point of E is of positive capacity even if the linear measure ofE vanishes.