| Summary: | Picard's theorem states that a non-constant function which is meromorphic in the complex plane C omits at most two values of the extended complex plane C*. A Picard set for a family of functions F is a subset E of the plane such that every transcendental f in F takes every value of C*, with at most two exceptions, infinitely often in C-E. If f is transcendental and meromorphic in the plane, then: (i) [Hayman and others] if N is a positive integer, f^Nf' takes all finite non-zero values infinitely often; (ii) [Hayman] either f takes every finite value infinitely often, or each derivative f^(k) takes every finite non-zero value infinitely often. We can seek analogues of Picard sets ie subsets E of the plane and an associated family of functions F, such that (for case (i)) f^Nf' takes all finite non-zero values infinitely often in C-E, for all f in F. Similarly for case (ii). In this thesis we improve or extend the results previously known, both for Picard sets proper and for the analogous cases (i) and (ii) mentioned above, when the family of functions F consists of meromorphic functions which have deficient poles (in the sense of Nevanlinna).
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