# Mappings by 1/z

Printable View

• Nov 15th 2008, 06:23 PM
mndi1105
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.
• Nov 15th 2008, 06:56 PM
ThePerfectHacker
Quote:

Originally Posted by mndi1105
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.
• Nov 16th 2008, 11:22 AM
shawsend
Quote:

Originally Posted by mndi1105
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. :)