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Re: Searching list for closest match to p

• To: mathgroup at smc.vnet.net
• Subject: [mg79418] Re: [mg79274] Searching list for closest match to p
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Thu, 26 Jul 2007 05:34:21 -0400 (EDT)
• References: <200707230741.DAA22149@smc.vnet.net>

chuck009 wrote:
> I'm working on an interesting theorem in Complex Analysis called Jentzsch's Theorem in which

the zeros of the partial sums of a Taylor series for an analytic 
function in the unit disk, all

converge to values on the unit disk.  So I choose a point on the unit 
disk, p=Cos[pi/3]+iSin[pi/3],

calculate the normal series for f[x]=Log[1+x] for n ranging from 1 to 
100, calculate the zeros, then

for each polynomial, search the zeros for the one closest to the point. 
  Here's my code.  I feel

the Table part is messy with the First[First[Position... construct in 
it.  Can anyone recommend a

more concise way of searching the zero lists and finding the one closest 
to p3?
> 
> Thanks,
> [...]

Corrected wording is that the collection of zeros of partial sums 
converge to all values on the unit disk (note juxtaposition vs. "all 
converge to").

I'd like someone to contribute an illustration of this result to the 
demonstrations.wolfram.com web site.* A possibility would be to have 
buttons to switch between a few possible functions with radius of 
convergence equal to one, a locator for the point on the circle, and 
have a slider that takes ever increasing partial sums of power series. 
It would then show those roots of the sums that get closer to the 
locator than roots of prior sums, perhaps making the color get of the 
points change as a function of proximity to the locator on the circle.

Daniel Lichtblau
Wolfram Research

* I'm prepared to be insufferable until I get one I like.

• References:
  • Searching list for closest match to p
    • From: chuck009 <dmilioto@comcast.com>