       Re: graphing x^2 + 4 on x, y, nad i

• To: mathgroup at smc.vnet.net
• Subject: [mg61890] Re: graphing x^2 + 4 on x, y, nad i
• From: Scott Hemphill <hemphill at hemphills.net>
• Date: Thu, 3 Nov 2005 04:59:08 -0500 (EST)
• References: <dka06q\$6v1\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```"G. Raymond Brown" <gbrown at runbox.com> writes:

> Thanks.  This provides an illuminating depiction of complex functions at
> both zeros and poles.

Another thing it can do is to depict the radius of convergence of a Taylor
series.

For example: 1/(1-z) = 1 + z + z^2 + ...

Plot4D[f_, args___] := Plot3D[{Abs[f], Hue[arg[f]/(2*Pi)]}, args]
arg[z_] := Arg[z]
arg[0. + 0.I] = 0  (* Arg[0.+0.I] is undefined; fix it *)

z=x+I*y
p[n_] := Plot4D[Sum[z^k,{k,0,n}], {x,-1,1}, {y,-1,1}, PlotRange->{0,20}]

The radius of convergence is one, which is clearly visible in the plot:

p

You can get Mathematica to plot 1/(1-z) by executing:

p[Infinity]

The sum doesn't actually converge outside the radius of convergence,
but Mathematica conveniently(?) ignores that fact.

Scott
--
Scott Hemphill	hemphill at alumni.caltech.edu
"This isn't flying.  This is falling, with style."  -- Buzz Lightyear

```

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