MHB Plotting Complex Functions: Does it Look Like a Riemann Surface?

[Dustinsfl]

I plotted the real and imaginary parts of a complex function \(z^{1/3}\). The two plots are similar to the Riemann surface is that correct?

 

[mathbalarka]

Recall that the Riemann surface of $w^3 = z$ is defined to be

$$\{(w_0, w_1, z_0, z_1) \in \Bbb R^4 \, :\, (w_0 + iw_1)^3 = z_0 + iz_1\}$$

While taking the real part you are really plotting

$$\{(x, y, z) \in \Bbb R^3 \, : \, z = \Re[(x+iy)^{1/3}]\}$$

So that's bound to be a section of the original Riemann surface, a bit deformed around the branches. Similar holds for the imaginary part.

So yes, those indeed look similar to the original Riemann surface.