# How to reproduce the Riemann Surface of Log[z]

I found this plot on Wikipedia:

which is the plot of the

multi-valued imaginary part of the complex logarithm function, which shows the branches.

So I tried to reproduce myself with the code:

myGradient = (Blend[{{0.345556, RGBColor[0.985, 0., 0.870436]}, {0,
RGBColor[0.359518, 0., 0.81767]}, {1,
RGBColor[0.443748, 1., 0.0305638]}, {0.572157,
RGBColor[0.985946, 0., 0.0269474]}, {0.79284,
RGBColor[1., 1., 0.041413]}}, #3] &);

Plot3D[{Im[Log[x + I y]], Im[Log[x + I y]] + 2 Pi,
Im[Log[x + I y]] + 4 Pi},
{x, -4, 4}, {y, -4, 4},
BoxRatios -> {1, 1, 1},
ImageSize -> Medium,
Mesh -> 25,
ColorFunction -> myGradient
]


which produces the output:

As you can notice I wasn't able to reproduce the curvy grid, which, instead, is the normal square grid of the Plot3D function. Further mor I have those strange white spaces which I would like to cover, to obtain an image the more similar to the one on Wikipedia.

Some ideas?

# EDIT:

I already read this question but by plotting with that code the Log[z] function I obtain this result:

which is clear not what I want.

• This may be helpful: mathematica.stackexchange.com/questions/31904/… – Jason B. Aug 9 '17 at 18:35
• so the surface is correct, you just want different mesh lines? The first question to ask is, what function of z do the mesh lines you want follow? The wikipedia page isn't clear on that. – Jason B. Aug 9 '17 at 18:45
• What if you added MeshFunctions -> {#1^2 + #2^2 &, ArcTan[#2, #1] &} as an option in the Plot3D call? That might not be quite it, but if you can figure out what the contour lines are, then you can adapt these functions to that. – march Aug 9 '17 at 18:45
• So, why did you not then try MeshFunctions with the real and imaginary parts of the logarithm? – J. M. is away Aug 10 '17 at 8:59
• If, as you say, you've figured it out, please do answer your own question. – J. M. is away Aug 10 '17 at 9:13

## 1 Answer

What I was looking for was to obtain the mesh lines of the mapping of a square grid on the complex plane $\mathbb{C}$ under the Log[z] transformation. So actually, to obtain the mesh lines in the image I posted (which is actually not so precise, as the lines coming from the origin don't seem to be straight lines), using the command MeshFunction, as suggested by march, who I thank, you only need to provide to it as functions respectively the imaginary and real part of the Log[z]:

Plot3D[{Im[Log[x + I y]], Im[Log[x + I y]] + 2 Pi,
Im[Log[x + I y]] - 2 Pi},
{x, -range, range}, {y, -range, range},
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
]