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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:

enter image description here

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]]

enter image description here

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1 Answer 1

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Try ParametricPlot3D:

myf[z_] := (2 + z) Log[2 + z] - 2 (1 + z) Log[1 + z] + z Log[z]; p1 = ParametricPlot3D[{Re[z], Im[z], Re[myf[z]]} /. z -> r Exp[I t], {r, 0, 3}, {t, -Pi, Pi}]; p2 = ParametricPlot3D[{Re[z], Im[z], Re[myf[z]] + 0.02} /.z -> r Exp[Pi I], {r, 0, 2}, PlotStyle -> {Thickness[0.005], White}]; Show[{p1, p2}]

enter image description here

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