Thread: sketch

sketch region where f(z)=z^3

0<r<2/3
0<theta<pi/4

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}]