Re: CMS: where are the zeroes ?

Tim Norfolk <timsn274@xxxxxxx> writes:

On Apr 26, 10:04=EF=BF=BDam, dan73 <fasttrac...@xxxxxxx> wrote:
As _n_ grows, the zeros are further and further away from 0. In fact,
each bound region of the complex plane contains only finitely many
suc=
h
zeros.
If a picture is worth a thousand words, how about an >animation?
Maple code:
with(plots):
display([seq](
pointplot(map([Re,Im],[fsolve(add
(z^j/j!,j=3D0..n),z,complex)]),
colour=3Dred,title=3D('n'=3Dn)),
n=3D1..50), insequence=3Dtrue);

<http://www.math.ubc.ca/~israel/exproots.gif>

What are the asymptotics of the curve on which the >roots lie?
--
Robert Israel isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics >http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

It appears that the points of these zeroes, @ any number
of iterations, that appear to the left of the y,-y axis
are for the most part parabolic.

At what point, for each n iterations, to the right of
the y,-y axis are they no longer parabolic?

Dan

Let the n-th partial sum be denoted $s_n(z)$, then Szego showed that
the limit points of the zeros of $s_n(nz)$ are precisely the points of
the curve $|ze^{1-z}|=3D1$, $|z| \le 1$. Later authors (Varga et al),
showed that the rate of convergence is $\frac{\ln n}{n}$, and that
convergence to $z=3D1$ is of order $\frac{1}{\sqrt{n}}$.

Thanks. Here's another animation, showing the scaled zeros and that
curve.

with(plots):
p0:= implicitplot(abs((x+I*y)*exp(1-x-I*y))=1,x=-1..1,y=-1..1,grid=[50,50],
gridrefine=2,crossingrefine=2,colour=blue):
display([seq](
display(p0, pointplot(map([Re,Im],[fsolve(add
(z^j/j!,j=0..n),z,complex)]/n),
colour=red,title=('n'=n))),
n=1..50), insequence=true);

<http://www.math.ubc.ca/~israel/exproots2.gif>
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

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