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arXiv · 2508.06724

Zeros of Harmonic Functions whose Caustic is a Non-Singular Image of an Epicycloid

Eli Sampson
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Abstract

Recent researchers have investigated how the zeros of certain families of complex harmonic functions change with a single parameter. Many leverage the well-behaved images of the critical curve and the harmonic analogue of the Argument Principle to prove zero-counting theorems. In this paper, we investigate the zeros of a family of harmonic functions for which the image of its critical curve is a non-singular linear image of an epicycloid. By analyzing this curve and using the harmonic analogue of the Argument Principle, we obtain a detailed zero-counting theorem for our family.

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Eli Sampson. 2025-08-08. Zeros of Harmonic Functions whose Caustic is a Non-Singular Image of an Epicycloid. https://arxiv.org/abs/2508.06724

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