The authors extend Pólya's shire theorem to hyperexponential functions f=(P/Q)exp(S/T), showing that normalized zero-counting measures of derivatives converge to a Voronoi edge measure augmented by weighted atoms at essential singularities, with explicit microscopic cluster laws.
Pólya,On the zeros of the derivatives of a function and its analytic character, Bull
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Zero asymptotics for successive derivatives of hyperexponential functions with finite essential singularities
The authors extend Pólya's shire theorem to hyperexponential functions f=(P/Q)exp(S/T), showing that normalized zero-counting measures of derivatives converge to a Voronoi edge measure augmented by weighted atoms at essential singularities, with explicit microscopic cluster laws.