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I'm an engineering-physics student taking a course in complex analysis, and it's a little frustrating, because I see all these connections to vector calculus over the reals (especially as applied to electromagnetics). For example, if we treat complex numbers like vectors, then the Cauchy-Riemann equations tell us that the derivative exists if the divergence and curl are both zero, which is equivalent to saying that our field is conservative, and there's no field sources/sinks. That these functions also satisfy Laplace's equation is also fairly obvious, since Laplace's equation describes a conservative field with no field sources.

Essentially, to what extent can we pretend the complex numbers are vectors, and bring over useful results from vector calculus? If this doesn't work, then what are the major flaws, or things to watch out for?

Edit: The reason I'm looking for analogs between vector analysis and complex analysis is because results in vector analysis usually have clear physical interpretations. I'm hoping to enhance my understanding of complex analysis through the similarities. Additionally, if I know a proof for certain proprerties of vector functions, I probably shouldn't focus my attention on the proof of an equivalent property in complex analysis. Instead, I'd prefer to focus attention on results that don't have a correspondence to vector analysis.

So I'm wondering, what are some results in complex analysis that don't have a similar result in vector analysis? Or even for results that are similar, what are the special peculiarities to complex analysis that make the results different?

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  • $\begingroup$ Sure, you could treat a complex-valued function just like a vector field, and apply the machinery of vector analysis perfectly well. The point, however, is that working with the field of complex numbers (as opposed to just the real plane) and with analytic functions (as opposed to just smooth vector fields) introduces an enormous amount of additional structure that makes life easier. So, sure, you could throw vector analysis at the world of complex analysis, but the world of complex analysis has so much more structure and is so much better behaved that it doesn't really help all that much. $\endgroup$ – Branimir Ćaćić Nov 1 '13 at 3:47
  • $\begingroup$ In the context of engineering-physics, we might use complex functions because they provide a more convenient representation for some physical phenomena, especially rotations, and sometimes complex solutions can suggest physical phenomena that otherwise wouldn't be obvious. They face the additional challenge though that there isn't always a good physical interpretation for complex numbers. If the additional structure of complex analysis makes life easier, can you provide an example? $\endgroup$ – David Nov 1 '13 at 4:01
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I think we do not really cheat in complex analysis. If I recall correctly, one of the popular proofs of Cauchy's integral formula is to use Green's theorem assuming the region is good enough. I think the reason you observe these things is because complex functions are harmonic. So tools in PDE and functional analysis naturally enters into the stage. Similarly you can prove an analytical function is infinitely many times differentiable by using maximal principle. But I think in real life the situation is the other way around, that we use complex functions as representations of "nicest" functions possible. And there is a geometric side which is not captured by the analytical nature of harmonic functions. For example a lot of complex functions are conformal, and understanding how it acts on the plane is a big deal to number theorists and analysts alike.

So I think you should regard complex analysis as a separate branch of analysis at this stage and know the textbook proofs really well, though you can always prove things on your own for enjoyment.

Edit:

Here is an somewhat related answer from the master:

https://mathoverflow.net/questions/3819/why-do-functions-in-complex-analysis-behave-so-well-as-opposed-to-functions-in/3830#3830

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    $\begingroup$ I think your answer is a good explanation of the similarities I see, but unfortunately doesn't satisfy my question. In particular, are there any results that exist vector analysis, but not in complex analysis, or vice versa? Keeping in mind I don't have the full background of complex analysis yet. $\endgroup$ – David Nov 1 '13 at 4:08
  • $\begingroup$ Well, any trivial example in 3 dimensions is not going to work in complex analysis. Do you think you have a nice proof of $e^{\pi i}=-1$ using "vector analysis"? $\endgroup$ – Bombyx mori Nov 1 '13 at 4:14
  • $\begingroup$ No, I obviously can't prove $e^{\pi i}=-1$ in vector analysis. The interpretation though, that a rotation of a vector by pi in a plane is the negative of the original vector, this isn't a new result (though it's definitely easier than a matrix representation for rotation). I agree looking for complex analogs of arbitrary 3d vector examples would be silly. $\endgroup$ – David Nov 1 '13 at 4:29
  • $\begingroup$ @David: Yes, the rotation interpretation is the right one. I think the most natural way to think why we need complex analysis is arguments involving the complex logarithm. The fact complex functions are multi-valued is what made them unique. $\endgroup$ – Bombyx mori Nov 1 '13 at 5:14

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