Riemann Surfaces

[SLD]

So, I was reading the Wikipedia article about Riemann Surfaces, and I’m very confused. How are they generated? They are projections of the Complex Plane into a third dimension, but exactly how is confusing to me.

so for example here is the Complex Plane for f(z)=Z



And so I compared it to my little app that does complex conformal mapping and this is what I got for the conformal map:



Possibly similar

But still different. So how do we calculate the shape of such surfaces?

 

[steve_bank]

What is the general definition of a 3d surface? What are the xyz parameters?

Scilab has 3d surface plot functions and examples. Free download.

 

[lpetrich]

A Riemann surface is a complex generalization of a 2D function curve: y = f(x) where both x and y are real numbers. It can be expanded as

yr = fr(xr,xi), yi = fi(xr,xi)

The usual way of graphing a Riemann surface is a colored 3D plot, where the x and y are the real and imaginary parts of the independent variable, the z is the complex magnitude of the function and the color is the complex phase of the function.