# Creating complex contour plots

I would like to create a contour plot of $e^z-\dfrac{z-1}{z+1}$ that looks like the picture on the left, but can only manage the one on the right. I have tried incrreasing the number of contours, but this doesn't have any effect. Am I missing something fundamental?

Show[Table[ContourPlot[{Im[E^(x + I y) - ((x + I y) - 1)/(x + I y) + 1] == k,
Re[E^(x + I y) - ((x + I y) - 1)/(x + I y) + 1] == k}, {x, -4, 4}, {y, -20, 20},
ContourStyle -> {Directive[{Red}], Directive[{Blue}]},
PlotPoints -> 100, Contours -> 20, AspectRatio -> Automatic], {k, -10, 10, 2}]]


The plot I am trying to recreate is from this website (specific PDF).

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The main thing you missed was that the picture on the left shows contours of the amplitude and phase, not the real and imaginary parts.

I couldn't get a good result for both sets of contours in one plot, so here I create the two plots separately and combine them with Show:

f[z_] := Exp[z] - (z - 1)/(z + 1)

Show[
ContourPlot[#1[f[x + I y]], {x, -4, 4}, {y, -20, 20},
ContourStyle -> #2, Contours -> #3,
AspectRatio -> Automatic, ContourShading -> None, PlotPoints -> 30] & @@@
{{Abs, Blue, Exp @ Range[-5, 5]},
{Arg, Red, Range[-Pi, Pi, Pi/5]}}
]


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Great - that you ! I just tried something different, but you are right, I did miss that - thank you again :) – martin Jun 8 '14 at 13:27