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I'm taking the complex variable course and I find the Wolfram Alpha website very helpful for me to understand the Riemann surface.

For example typing "Riemann surface z^(1/2)" I get nice plots

enter image description here

but when I try (z+1)^(1/2)+(z-1)^(1/2), I get something like this:

enter image description here

As I understand, the branch cut happens at $(-\infty,-1)$ and $(1,\infty)$, then why there are cuts between -1 and 1 in the plot? Did I make a stupid mistake?

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  • $\begingroup$ Isn't the entire point of Riemann surfaces that there is no branch cut because there is an entire surface - and the point of a branch cut to change a surface into the graph of a function? $\endgroup$ – Carl Mummert Sep 9 '13 at 23:24
  • $\begingroup$ @CarlMummert I'm sorry that I don't see your point. I agree that the Riemann surface is an entire surface, and branch cut is the place that Riemann sheets cross, but how does that relate to the question? $\endgroup$ – xslittlegrass Sep 9 '13 at 23:56
  • $\begingroup$ I can't make heads or tails of that surface. Did you compare it to the surface of the product $\sqrt{z-1}\sqrt{z+1}$ ? That one should really be a cut between $-1,1$. When I did it on wolfram alpha it seems to be different. $\endgroup$ – Evan Sep 10 '13 at 0:06

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