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 \[ Z_{\theta_{1},\theta_{2}} := \left\{z \in {\bf C} : \theta_{1} \leq \arg z \leq \theta_{2}\right\}. \] If $\#\left(\left\{z_{k,n}\right\}_{k=1}^{n} \bigcap Z_{\theta_{1},\theta_{2}}\right)$ represents the number of zeros of $s_{n}(z)$ in the sector $Z_{\theta_{1},\theta_{2}}$, then Szegő showed in 1924 that \[ \lim_{n\rightarrow \infty} \frac{\# \left( \{ z_{k,n}\}^{n}_{k=1} \bigcap Z_{\theta_{1}, \theta_{2}} \right) }{n} = \frac{\phi_{2} - \phi_{1} }{2\pi}, \] where $\phi_{1}$ and $\phi_{2}$ are defined in the text. The associated discrepancy function is defined by \[ {\rm disc}_{n} (\theta_{1}, \theta_{2} ) := \# \left( \{ z_{k,n}\}^{n}_{k=1} \bigcap Z_{\theta_{1}, \theta_{2}} \right) - n\left( \frac{\phi_{2} - \phi_{1}}{2\pi} \right). \] One of our new results shows, for any $\theta_{1}$ with $0 < \theta_{1} < \pi$, that \[ {\rm disc}_{n} (\theta_{1},2\pi - \theta_{1}) \sim K\log n, ~~{\rm as}~~ ~{n\rightarrow \infty}, \] where $K$ is a positive constant, depending only on $\theta_{1}$. Also new in this paper is a study of the cyclical nature of ${\rm disc}_{n}(\theta_{1}, \theta_{2})$, 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 $\phi_{1}$. 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