Working Paper Article Version 1 This version is not peer-reviewed

# Taming the Natural Boundary of Centered Polygonal Lacunary Functions: Restriction to the Symmetry Angle Space

Version 1 : Received: 19 February 2020 / Approved: 19 February 2020 / Online: 19 February 2020 (11:43:50 CET)

How to cite: Mork, L.; Sullivan, K.; Ulness, D.J. Taming the Natural Boundary of Centered Polygonal Lacunary Functions: Restriction to the Symmetry Angle Space. Preprints 2020, 2020020276 Mork, L.; Sullivan, K.; Ulness, D.J. Taming the Natural Boundary of Centered Polygonal Lacunary Functions: Restriction to the Symmetry Angle Space. Preprints 2020, 2020020276

## Abstract

This work investigates centered polygonal lacunary functions restricted from the unit disk onto symmetry angle space which is defined by the symmetry angles of a given centered polygonal lacunary function. This restriction allows for one to consider only the p-sequences of the centered polygonal lacunary functions which are bounded, but not convergent, at the natural boundary. The periodicity of the $p$-sequences naturally gives rise to a convergent subsequence, which can be used as a grounds for decomposition of the restricted centered polygonal lacunary functions. A mapping of the unit disk to the sphere allows for the study of the line integrals of restricted centered polygonal that includes analytic progress towards closed form representations. Obvious closures of the domain obtained from the spherical map lead to four distinct topological spaces of the "broom topology'' type.

## Subject Areas

lacunary function; gap function; centered polygonal numbers; Natural boundary; singularities; broom topology

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