Plotting Complex functions in Mathematica

[Storm Butler]

In a few of my books on Complex variables they show how you can look at a complex function as essentially a mapping from what plane to another.

Does anyone know if there would be a way to have mathematica plot how a complex function would transform one plane into the other?

Thanks for any help.

 

[jackmell]

In a few of my books on Complex variables they show how you can look at a complex function as essentially a mapping from what plane to another.

Does anyone know if there would be a way to have mathematica plot how a complex function would transform one plane into the other?

Thanks for any help.

You can plot a parametric region. For example, the mapping $e^z\to w$ maps the square region in the z-plane to the washer region in the w-plane:

w[z_] := Exp[z]; 
p1 = ParametricPlot[{x, y}, {x, 1/10, 1}, {y, -Pi, Pi}, AspectRatio -> 1]; 
p2 = ParametricPlot[{Re[w[z]], Im[w[z]]} /. z -> x + I*y, {x, 1/10, 1}, {y, -Pi, Pi}, 
    PlotRange -> All]; 
myarrow = Show[Graphics[{{Arrow[{{-0.5, 0}, {0.5, 0}}]}, 
      Text[Style["w[z]=\!\(\*SuperscriptBox[\(E\), \(z\)]\)", 20], {0, 0.1}]}]]; 
GraphicsGrid[{{p1, myarrow, p2}}]

Ok, now modify my code to map the annulus $1\leq r\leq 5$ in the z-plane under the transformation $1/z\to w$ into the w-plane.