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Limit Curves for Zeros of Sections of Exponential Integrals

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Abstract

We are interested in studying the asymptotic behavior of the zeros of partial sums of power series for a family of entire functions defined by exponential integrals. The zeros grow on the order of \(O(n)\), and after rescaling, we explicitly calculate their limit curve. We find that the rate at which the zeros approach the curve depends on the order of the singularities/zeros of the integrand in the exponential integrals. As an application of our findings, we derive results concerning the zeros of partial sums of power series for Bessel functions of the first kind.

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Acknowledgments

The author would like to thank the referees for their valuable questions and comments.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Dalhousie University, Halifax, NS , B3H 4J5, Canada

    Antonio R. Vargas

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  1. Antonio R. Vargas
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Correspondence to Antonio R. Vargas.

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Communicated by Edward B. Saff.

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Vargas, A.R. Limit Curves for Zeros of Sections of Exponential Integrals. Constr Approx 40, 219–239 (2014). https://doi.org/10.1007/s00365-014-9241-7

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