We use a nonstandard model of analysis to study two main topicsin complex analysis.
UNIFORM CONTINUITY AND RATES OF GROWTH OF MEROMORPHIC FUNCTIONSis a unified nonstandard approach to severaltheories; the Julia-Milloux theorem and Julia exceptional functions,Yosida's class (A), normal meromorphic functions, and Gavrilov'sWp classes. All of these theories are reduced to the study of uniformcontinuity in an appropriate metric by means of S-continuity in thenonstandard model (which was introduced by A. Robinson).
The connection with the classical Picard theorem is madethrough a generalization of a result of A. Robinson on S-continuous*-holomorphic functions.
S-continuity offers considerable simplifications over the standardsequential approach and permits a new characterization of these growthrequirements.
BOUNDED ANALYTIC FUNCTIONS AS THE DUAL OF A BANACH SPACE is a nonstandard approach to the pre-dual Banachspace for H∞(D) which was introduced by Rubel and Shields.
A new characterization of the pre-dual by means of thenonstandard hull of a space of contour integrals infinitesimally near theboundary of an arbitrary region is given.
A new characterization of the strict topology is given in termsof the infinitesimal relation: "h b k provided ||h-k|| is finite andh(z) ≈ k(z) for z∈(*D)".
A new proof of the noncoincidence of the strict and Mackeytopologies is given in the case of a smooth finitely connected region.The idea of the proof is that the infinitesimal relation: "h γ k provided||h-k|| is finite and h(z) ≈ k(z) on nearly all of the boundary", givesrise to a compatible topology finer than the strict topology.
