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Suppose I have only real number problems, where I need to find solutions. By what means could knowledge about complex numbers be useful?

Of course, the obviously applications are:

  • contour integration
  • understand radius of convergence of power series
  • algebra with $\exp(ix)$ instead of $\sin(x)$

No need to elaborate on these ones :) I'd be interested in some more suggestions!

In a way this question is asking how to show the advantage of complex numbers for real number mathematics of (scientifc) everyday problems. Ideally these examples should provide a considerable insight and not just reformulation.

EDIT: These examples are the most real world I could come up with. I could imagine an engineer doing work that leads to some real world product in a few months, might need integrals or sine/cosine. Basically I'm looking for a examples that can be shown to a large audience of laymen for the work they already do. Examples like quantum mechanics are hard to justify, because due to many-particle problems QM rarely makes any useful predictions (where experiments aren't needed anyway). Anything closer to application?

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    possible duplicate of [Interesting results easily achieved using complex numbers](http://math.stackexchange.com/questions/4961/interesting-results-easily-achieved-using-complex-numbers)2012-11-25
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    I don't think this is a duplicate of the above question. In the present one the OP wants to know *real* (as in the real world) applications of complex numbers, in the linked one those were more mathematical applications.2012-11-25
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    Yes, as real as mathematics can get. Maybe something that could excite an engineer. Or even more real :)2012-11-25
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    To begin with, there're lots of mathematical stuff that is waaaaay simpler using complex analysis than trying just real one, and I believe there're things that cannot be done, or that at least we don't know how, without complex analysis2012-11-25
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    You should edit your question, then. Change "real number" to "real world", so people won't mistake it for the real numbers.2012-11-25
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    Control theory! Fluid dynamics! Differential equations! Electrical engineering! Signal processing! Quantum mechanics! http://en.wikipedia.org/wiki/Complex_number#Applications2012-11-25
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    This question is quite ambiguously phrased: all three applications listed in it belong to pure mathematics (and two answers posted so far address some aspects of *this*) but afterwards the OP claims to be interested in "real world" applications, where "real world" seems to be more or less equivalent to "useful to an engineer" (and @Rahul's comment answers *that* beautifully). Please make up your mind.2012-11-25
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    They're used incessantly in electrical engineering.2012-11-25
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    @did: These examples are the most real-world I could come up with. An engineer might calculate with sine/cosine or might be interested in the value of an integral for some engineering that leads to a real world product. If you know a more real world example feel free to answer. Don't complain if you don't see the connection.2012-11-26
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    ?? *Complain*? Well... If ever I had fancied answering the question, your last comment is a quite effective deterrent. (Update: upon reading your comment, I was vaguely wondering when I had previously met this tone on the site... and [behold!](http://math.stackexchange.com/a/187796))2012-11-26
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    Maybe this can also help: http://mathforum.org/library/drmath/view/53879.html2012-11-26

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