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

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The discussion confirms that the plots of the real and imaginary parts of the complex function \(z^{1/3}\) resemble the Riemann surface defined by \(w^3 = z\). It explains that plotting the real part corresponds to a section of the Riemann surface, albeit deformed around branch points. The same applies to the imaginary part, reinforcing the similarity to the original Riemann surface. Overall, the analysis supports the conclusion that the plotted functions visually align with the characteristics of the Riemann surface. The findings highlight the connection between complex function plots and Riemann surfaces in mathematical visualization.
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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?

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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.
 

Thread 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
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