sketch region where f(z)=z^3

0<r<2/3

0<theta<pi/4

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- Feb 1st 2010, 11:33 PMstumped765sketch
sketch region where f(z)=z^3

0<r<2/3

0<theta<pi/4 - Feb 2nd 2010, 05:10 AMshawsend
We have then consider and just for now, let r=1, and t go from 0 to then wouldn't that be an arc of radius one but going from 0 to ? Same dif for all the other arcs as the radius goes from zero to then we have the piece with radius 0 to going from 0 to as shown in the plot below. Here's the Mathematica code to draw f(z) in some region defined by the region parameters. Try and find a machine with Mathematica and experiment with it.

Code:`mapregion[f_, region_, xrange_, yrange_] :=`

Module[{real, imag, rplot, transform, newplot},

real = ComplexExpand[Re[f /. z -> x + I*y]];

imag = ComplexExpand[Im[f /. z -> x + I*y]];

rplot = RegionPlot[region, {x, xrange[[1]], xrange[[2]]},

{y, yrange[[1]], yrange[[2]]}, PlotPoints -> 75,

AxesLabel -> {Style["x", 20], Style["y", 20]}, Frame -> None,

Axes -> True, AspectRatio -> 1, PlotRange -> {xrange, yrange}];

transform = rplot /. GraphicsComplex[pnts_, data__] :>

GraphicsComplex[({real, imag} /. {x -> #1[[1]], y -> #1[[2]]} & ) /@

pnts, data]; transform = transform /. RGBColor[x_, y_, z_] ->

RGBColor[1, 0, 0]; newplot = Show[{transform},

PlotRange -> {xrange, yrange}, AxesLabel -> {Style["u", 20],

Style["v", 20]}, Frame -> None, Axes -> True];

GraphicsArray[{{rplot, newplot}}]];

mapregion[z^3, Inequality[0, Less, Abs[x + I*y], LessEqual, 2/3] &&

0 <= Arg[x + I*y] <= Pi/4, {-1, 1}, {-1, 1}]