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

Dustinsfl
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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?

twlfDg3.png


qTfSwPj.png
 
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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.
 
I posted this question on math-stackexchange but apparently I asked something stupid and I was downvoted. I still don't have an answer to my question so I hope someone in here can help me or at least explain me why I am asking something stupid. I started studying Complex Analysis and came upon the following theorem which is a direct consequence of the Cauchy-Goursat theorem: Let ##f:D\to\mathbb{C}## be an anlytic function over a simply connected region ##D##. If ##a## and ##z## are part of...
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