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Analytic Functions with an IrregularLinearly Measurable Set of Singular Points

Published online by Cambridge University Press:  20 November 2018

I. E. Glover
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V. V. Golubev, in his study [6], has constructed, by using definite integrals, various examples of analytic functions having a perfect nowhere-dense set of singular points. These functions were shown to be single-valued with a bounded imaginary part. In attempting to extend his work to the problem of constructing analytic functions having perfect, nowhere-dense singular sets under quite general conditions, he posed the following question: Given an arbitrary, perfect, nowhere-dense point-set E of positive measure in the complex plane, is it possible to construct, by passing a Jordan curve through E and by using definite integrals, an example of a single-valued analytic function, which has E as its singular set, with its imaginary part bounded.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 4 , 1952 , pp. 424 - 435
Copyright
Copyright © Canadian Mathematical Society 1952

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