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Math Help - Mappings by 1/z

  1. #1
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    Mappings by 1/z

    a) Find the image of the infinite strip 0 < y < 1/(2c) under the transformation w=1/z. Sketch the strip and its image.


    b) Find the image of the quadrant x>1, y>0 under the transformation w=1/z.
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  2. #2
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    Quote Originally Posted by mndi1105 View Post
    a) Find the image of the infinite strip 0 < y < 1/(2c) under the transformation w=1/z. Sketch the strip and its image.


    b) Find the image of the quadrant x>1, y>0 under the transformation w=1/z.
    Try seeing where the boundary curves gets mapped to. Remember 1/z maps lines and circles into lines and circles. That should be enough infromation.
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  3. #3
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    Quote Originally Posted by mndi1105 View Post
    a) Find the image of the infinite strip 0 < y < 1/(2c) under the transformation w=1/z. Sketch the strip and its image.


    b) Find the image of the quadrant x>1, y>0 under the transformation w=1/z.
    Here's a Mathematica routine to plot the image of complex functions over user-defined regions. It has some limitations. I plotted the two regions below. mapregion takes the function, the region (such as 0<=y<=.5 && -10<=x<=10), and the x and y plot ranges ({-8,8}, {-8,8}). What's that hole doing in the plots anyway? Is is my mistake? How would you call mapregion to plot the region for the second problem?

    Code:
    mapregion[1/z, 0 <= y <=.5 && -10 <= x <= 10, 
              {-8, 8}, {-8, 8}]
    
    mapregion[1/z, 0.1 <= y <= 1 && -10 <= x <= 10, 
              {-8, 8}, {-8, 8}]
    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, -5, 5}, 
           {y, -2, 2}, PlotPoints -> 75, 
           AxesLabel -> {Style["x", 20], 
             Style["y", 20]}, Frame -> None, 
           Axes -> True]; transform = 
          rplot /. GraphicsComplex[pnts_, 
             data__] :> GraphicsComplex[
             ({real, imag} /. {x -> #1[[1]], 
                 y -> #1[[2]]} & ) /@ pnts, 
             data]; newplot = Show[transform, 
           PlotRange -> {xrange, yrange}, 
           AxesLabel -> {Style["u", 20], 
             Style["v", 20]}, Frame -> None, 
           Axes -> True]; GraphicsArray[
          {{rplot, newplot}}]];
    Oh yea, you need to try and do these mapping by hand first before you start relying on Mathematica right? Mathematica is like a 50 caliber rifle: Fun to use but you need to be careful using it.
    Attached Thumbnails Attached Thumbnails Mappings by 1/z-map1.jpg   Mappings by 1/z-map2.jpg  
    Last edited by shawsend; November 16th 2008 at 11:35 AM.
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