Volume 30, pp. 128-143, 2008.

The dynamical motion of the zeros of the partial sums of $e^z$, and its relationship to discrepancy theory

Richard S. Varga, Amos J. Carpenter, and Bryan W. Lewis

Dedicated to Edward B. Saff on his 64th birthday, January 2, 2008.

Abstract

With $s_n(z):=\sum _{k=0}^n{z}^k/k!$ denoting the $n$-th partial sum of $e^z$, let its zeros be denoted by ${\left\{z_{k,n}\right\}}_{k=1}^n$ for any positive integer $n$. If ${\theta }_1$ and ${\theta }_2$ are any angles with $0<{\theta }_1<{\theta }_2<2\pi$, let $Z_{{\theta }_1,{\theta }_2}$ be the associated sector, in the z-plane, defined by $$Z_{{\theta }_1,{\theta }_2}:=\{z\in \mathbf{C}:{\theta }_1\le \arg z\le {\theta }_2\}.$$ If $\mathrm{\#}({\left\{z_{k,n}\right\}}_{k=1}^n\bigcap Z_{{\theta }_1,{\theta }_2})$ represents the number of zeros of $s_n(z)$ in the sector $Z_{{\theta }_1,{\theta }_2}$, then Szegő showed in 1924 that $$\underset{n\to \mathrm{\infty }}{lim}\frac{\mathrm{\#}(\{z_{k,n}{\}}_{k=1}^n\bigcap Z_{{\theta }_1,{\theta }_2})}{n}=\frac{{\varphi }_2-{\varphi }_1}{2\pi },$$ where ${\varphi }_1$ and ${\varphi }_2$ are defined in the text. The associated discrepancy function is defined by $${\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}_n({\theta }_1,{\theta }_2):=\mathrm{\#}(\{z_{k,n}{\}}_{k=1}^n\bigcap Z_{{\theta }_1,{\theta }_2})-n\left(\frac{{\varphi }_2-{\varphi }_1}{2\pi }\right).$$ One of our new results shows, for any ${\theta }_1$ with $0<{\theta }_1<\pi$, that $${\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}_n({\theta }_1,2\pi -{\theta }_1)\sim K\log n,\mathrm{a}\mathrm{s}n\to \mathrm{\infty },$$ where $K$ is a positive constant, depending only on ${\theta }_1$. Also new in this paper is a study of the cyclical nature of ${\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}_n({\theta }_1,{\theta }_2)$, as a function of $n$, when $0<{\theta }_1<\pi$ and ${\theta }_2=2\pi -{\theta }_1$. An upper bound for the approximate cycle length, in this case, is determined in terms of ${\varphi }_1$. All this can be viewed in our Interactive Supplement, which shows the dynamical motion of the (normalized) zeros of the partial sums of $e^z$ and their associated discrepancies.

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Key words

partial sums of $e^z$, Szegő curve, discrepancy function

AMS subject classifications

30C15, 30E15

Additional resources for this document

Interactive Supplement