# visualizing branch cut of a complex function

I'm working in the plotting the branch cut of a complex function, namely:

$$w(z) = (2+z) \ln(2+z) - 2(1+z) \ln(1+z) + z \ln z$$.

To do so, I have tried this:

Plot3D[Re[(2 + (x + I y)) Log[2 + (x + I y)] -
2 (1 + (x + I y)) Log[
1 + (x + I y)] + (x + I y) Log[(x + I y)]], {x, -3, 3}, {y, -3,
3}, PlotRange -> All]


There is a white blank on the plot from $$-3$$ to $$-2$$ which is NOT due to the branch cut, and shouldn't be there:

How to remove this "extra white line"? (I'd like to have the branch cut as a white line, that is to keep the white line from $$−2$$ to $$0$$. How can I remove the white line from $$−3$$ to $$-2$$ which is not a branch cut and I don't know why Mathematica produces it.)

Edit:

The contour plot also gives the same bad result:

With[{z = x + I y},
ContourPlot[
Re[(2 + z) Log[2 + z] - 2 (1 + z) Log[1 + z] + z Log[z]], {x, -3,
3}, {y, -3, 3}, Contours -> Range[-4, 2, .1],
ColorFunction -> (ColorData["Rainbow"][Rescale[#, {-2, 1}]] &),
ColorFunctionScaling -> False, PlotRange -> All]]


myf[z_] := (2 + z) Log[2 + z] - 2 (1 + z) Log[1 + z]