Re: Searching list for closest match to p
- To: mathgroup at smc.vnet.net
- Subject: [mg79546] Re: Searching list for closest match to p
- From: chuck009 <dmilioto at comcast.com>
- Date: Sun, 29 Jul 2007 00:06:19 -0400 (EDT)
Hello Peter, I think your example of f[z]=x/(x-2)^2 has a radius of convergence of 2 and I think the theorem may only apply to a unit disk of convergence. I edited your code and note that the points "seem" to be tending to p3=2e^{pi/3}. I'm really not clear at all about this theorem and lots of other things in Complex Analysis.
p3color = Red;
circleColor = Blue;
p3 = 2Exp[I*Pi/3];
f[x_] = x/(x - 2)^2;
s100 = SeriesCoefficient[Series[f[x], {x, 0, 100}], #]x^# & /@ Range[100];
p3vals = Through[{Re, Im}[#]] & @@@ Table[zlist =
x /. NSolve[Total[Take[s100, nval]] == 0];
Pick[zlist, #, Min[#]] &[Abs[zlist - p3]], {nval, 100}];
lp3 = ListPlot[p3vals, PlotRange -> {{0, 2.2}, {0, 2.2}},
AspectRatio -> Automatic, Epilog -> {
circleColor, Circle[{0,
0}, 2, {0, Pi/2}], p3color, PointSize[.03], Point[Through[{Re,
Im}[p3]]]}]
> I didn't try to prove this (too lazy) but I've got
> t an counterexample
> for your version:
> f(x)=x/(x-2)^2 is analytic on the unit disc, but: