How are Riemann surfaces graphed

[saminator910]

I am creating a python application to graph riemann surfaces, from the wikipedia article it says the functions are graphed as x real part of the complex domain, y, imaginary part of the complex domain, z, the real part of f(z), but how does one represent the imaginary part with the color, would you base it on the angle of the complex f(z) vector?

 

[jackmell]

Those plots are the real or imaginary components of the covering of a Riemann surface over the z-plane and aren't really Riemann surfaces in the classical sense. The colors are irrelevant. Not hard to start drawing them. Take for example $f(z)=\sqrt{z}$. That is a 2-covering of the comple plane since it has two values for each value of z. I could draw each covering separately and then just plot them together to form the 2-covering represented by the function and Wikipedia. To draw the first covering say in Mathematica since that's what I use, I'd draw it parametrically as (pseudo-code):

p1=ParametricPlot3D[{Re[z],Im[z],Re[sqrt[z]]}/.z->r e^{it},{r,0,1},{t,-pi,pi}]

That is, let z=re^{it}, then for the coordiate point (Re(z),Im(z)), draw the real part of sqrt(z) and then do that in a disc of radius 1.

That's one covering. Now since the other covering is just $-\sqrt{z}$, to plot that one, I'd use:

p2=ParametricPlot3D[{Re[z],Im[z],-sqrt[z]}/.z->r e^{it},{r,0,1},{t,-pi,pi}]

Now just show them together:

Show[{p1,p2}]

That then would give you the 2-covering of that function. It's the simplest way to plot root functions and there are more sophisticated ways to plot more complicated functions using differential equations.