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Ever since I learned about complex valued functions I've been wondering if there was a better visualization for them. Obviously we can't visualize four dimensions, but I was wondering if it would be useful to model them as vector fields. After all, a complex valued function is essentially a mapping from $\mathbb{R}^2$ to $\mathbb{R}^2$. Are there useful aspects of seeing it this way, and what benefits do other visualizations have over this?

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    $\begingroup$ It would be hard to visualize the branch cuts in this way. $\endgroup$ – mastrok Jun 16 '16 at 4:44
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Yes, for some purposes this can be useful, as described (much better than I could do here) in Chapters 10 and 11 in the book Visual Complex Analysis by Tristan Needham.

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