Re: Zeroes of exp...
From: Robert Israel (israel_at_math.ubc.ca)
Date: 01/11/05
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Date: 11 Jan 2005 00:57:24 GMT
In article <de044e77.0501101001.209ed595@posting.google.com>,
tetrahedron <jarynth@yahoo.com> wrote:
>... don't exist. But each approximation of exp given by the n-th
>partial sum of the exponential series centered at 0 (considering 1 as
>the 0-th) does have some zeroes in the complex plane, actually n. When
>I evaluated and plotted such roots (as far as numerical evaluation
>took me) I found out they arrange in a nice pattern. I was wondering:
>1) Do the roots of the n-th partial sums lie on some particular curve?
>How can this be proven/disproven?
I found this in MathSciNet:
MR1113918 (92m:33003)
Carpenter, A. J.(1-BUTL); Varga, R. S.(1-KNTS-CM); Waldvogel,
J.(CH-ETHZ-AM)
Asymptotics for the zeros of the partial sums of $e\sp z$. I.
Proceedings of the U.S.-Western Europe Regional Conference on Padé
Approximants and Related Topics (Boulder, CO, 1988).
Rocky Mountain J. Math. 21 (1991), no. 1, 99--120.
33B10 (30E15 41A60)
References: 0 Reference Citations: 2 Review Citations: 2
Let $s_n(z)\coloneq\sum^n_{j=0}z^j/j!$, $n\geq 1$. In this well-written
paper the authors improve a result of G. Szegö \ref[Sitzungsber. Berlin
Math. Ges. 23 (1924), 50--64], that all zeros of $s_n(nz)$, as
$n\to\infty$, accumulate on the curve $D_\infty\coloneq\{z\in\bold C\colon
|ze^{1-z}|=1$, $|z|\leq 1\}$. If $Z_n$ is the set of zeros of $s_n(nz)$,
$C_\delta\coloneq\{z\in\bold C\colon |z-1|<\delta\}$, $0<\delta\leq 1$,
and if for $D\subset\bold C$ one defines the distance ${\rm
dist}(Z_n,D)\coloneq\max_{z\in Z_n}\,{\rm dist}(z,D)$, then the following
are proved: $$\liminf_{n\to\infty}n^{1/2}\,{\rm dist}(Z_n,D_\infty)>0,$$
$${\rm dist}(Z_n\sbs C_\delta,D_\infty)=O(n^{-1}\log n)\quad{\rm and}$$
$$\liminf_{n\to\infty}(n/\log n)\,{\rm dist}(Z_n\sbs
C_\delta,D_\infty)>0$$ for each $0<\delta\leq 1$. If $a_n\coloneq
n!n^{-n}e^n$ and $D_n\coloneq\{z\in\bold C\colon\
|ze^{1-z}|^n=a_n|1-z|/|z|$, $|z|\leq 1$, $\cos(\arg z)\leq 1-2/n\}$, then
${\rm dist}(Z_n\sbs C_\delta,D_n)=O(n^{-2})$ and
$\liminf_{n\to\infty}n^2\,{\rm dist}(Z_n\sbs C_\delta,D_n)>0$ for each
$0<\delta\leq 1$.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
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