# Visualizing Riemann surface (two branches) of logarithm

I'm trying to plot two branches of the complex multi-valued function: (In a previous post - linked above-, Mathematica found the branch cut of the following function between -1 and 0)

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

So, to plot the real parts of the two sheets, we have:

Plot3D[{Re[1 - (x + I y) (Log[1 + x + I y] - Log[x + I y])],
Re[1 - (x + I y) (Log[1 + x + I y] - Log[x + I y] -
I 2 Pi)]}, {x, -2, 2}, {y, -3, 3}, BoxRatios -> {1, 1, 1.5},
PlotRange -> All, PlotPoints -> 50, Mesh -> 30,
MeshFunctions -> {Im[Log[#1 + I #2]] &, Re[Log[#1 + I #2]] &},
ImageSize -> Large, ColorFunction -> mycolor]


which as expected the two sheets are connected through the branch cut between -1 and 0 on the x-axis:

However, the pictures shows another branch cut between -2 and -1. How come?

• @MariuszIwaniuk In that question, I needed to know if the Mathematica understands the branch cut of the logarithm. Here, I'm interested in its Riemann surface. – user67023 Aug 18 '19 at 13:18

f[z_, n_] := 1 - z (Log[Abs[(1 + z) z]] + I (2 n Pi + Arg[(1 + z)/z]))