To plot branch cut of logarithm

Question

I like to see the branch cut of the function:

$$1 - z \ln[(1+z)/z].$$

If I plot it in the complex plane:

Plot3D[Re[1 - (x + I y) Log[(1 + x + I y)/(x + I y)]], {x, -2, 
2}, {y, -2, 2}]

The result is:

which shows the brach cut correctly between -1 and 0. How can I get rid of the hole in the picture and have a smooth line as a branch cut rather than a white line discontinuity?

Also the same for contour plot:

With[{z = x + I y}, 
ContourPlot[Re[1 - z Log[(1 + z)/(z)]], {x, -2, 2}, {y, -2, 2}, 
Contours -> 40]]

Answer 1

Change "Rainbow" with any ColorScheme you prefer, and the rescaling values {-2,1} to obtain different scaling.

With[{z = x + I y}, 
 ContourPlot[Re[1 - z Log[(1 + z)/(z)]], {x, -2, 2}, {y, -2, 2}, 
  Contours -> Range[-4, 2, .1], 
  ColorFunction -> (ColorData["Rainbow"][Rescale[#, {-2, 1}]] &),
  ColorFunctionScaling -> False, PlotRange -> All]]

Answer 2

Note that you can also use the new (as of Version 12) ComplexPlot function, too:

ComplexPlot[1 - z Log[(1 + z)/z], {z, -2 - 2 I, 2 + 2 I}, Mesh -> 10, 
   MeshFunctions -> {Re[#2] &, Im[#2] &}]

Or the 3D version:

ComplexPlot3D[1 - z Log[(1 + z)/z], {z, -2 - 2 I, 2 + 2 I}, 
   Mesh -> 10, PlotRange -> All]