ComplexAnalysis is the study of functions of complex variables (ComplexNumbers) in the same way that RealAnalysis is the study of functions of real variables. (helpful, no?). It focuses mainly on HolomorphicFunctions (differentiable complex functions), and on differentiation and integration of these functions.

It seems intuitively obvious that RealAnalysis has many application, due to the obvious mappings between real functions and physical systems, and also because RealAnalysis came about as a tool to study physical phenomenon. But ComplexAnalysis, despite its strangeness and the fact it was developed with purely mathematical motivations, also has a wealth of surprising physical applications. It is used in signal processing, electrical engineering, engineering, and in physics in the study of fluid flow. Among many other things. Perhaps its strangest application is in QuantumMechanics, where complex numbers are used to represent (roughly) the probabilities of different states of a system.

See HestenesOerstedMedalLecture for a different view on complex numbers in QuantumMechanics.

''The above is "damning with faint praise"; the surprising thing is that someone with an appreciation of the subject would laud it so little. It's not that it has many applications, it's closer to the case that it is the default tool of choice for dealing with physical phenomenon. RealAnalysis is '''not''' (in general) a more natural tool for the physical world, and complex numbers are not surprising to find in physics; they are everywhere. Real numbers are often what measuring devices are able to measure, but nonetheless are typically not as natural as a theoretical model as are things in the complex number domain.''

The fact that complex numbers are everywhere in physics is only "not surprising" once you've stopped being surprised by it. When you're learning, especially if you're coming from a maths perspective, it seems very strange indeed that these things come up in so many unrelated places.

And my comparison with RealAnalysis was to the relative ''apparent'' usefulness of the two subjects, not the actual amount they are used by working physicists, and physics students. It seems to me that, to someone who was coming to physics for the first time, RealAnalysis ''does'' seem obviously more useful, just because it deals with RealNumbers, something we intuitively map to all kinds of phenomena, as opposed to ComplexNumbers, that don't so obviously map to physical phenomena. -- RobbieCarlton

''But surely the surprise is about complex numbers themselves, not ComplexAnalysis, and is highly parallel to the "unintuitiveness" that the ancients felt for "zero as a number" (which hindered, e.g., EuclidOfAlexandria's description of the GCD algorithm; see Knuth), for negative numbers, for irrational numbers, and of course imaginary numbers - the intellectual opposition for the latter two being memorialized in their very names; there's nothing irrational nor imaginary about them, in the everyday senses of the words. Not to mention the even bigger issues with non-finite "numbers" (settled by NonStandardAnalysis; indirect work-arounds no longer essential) as critiqued by, e.g., Zeno and Bishop Berkley.''

''Once one gets past the notion that naive intuitions are the ultimate judge, and one retrains one's intuition to accept all these abstractions of the concept of "number", after that, surely real world applications of such concepts are not surprising at all, well before even studying ComplexAnalysis.''

''In a real sense, perhaps it is much more surprising that RealAnalysis is not exactly a subset of ComplexAnalysis, although it approximates one in some ways.''

* Doug (for I think it was you) - I agree with you completely in your assertions that ComplexAnalysis is in fact more useful, and that Real Analysis is largely a subset - a special case. I have to agree with the original author, though, about perceptions. Students can clearly see how "real" numbers are obviously useful, corresponding directly to measurements and their own experiences. Complex numbers are more abstract, and although when they become your friends it is clear that they are the right things to work with, nevertheless the initial perception tends to be one of confusion, and the initial response is "How will I ever use these?". Initial impressions matter, complex numbers are more abstract, and some people struggle with that.
* ''I suppose I may be in "ViolentAgreement" without having noticed, since I certainly can't disagree that these things are "only not surprising once you've stopped being surprised by it". Possibly my experience is fairly unusual, in that I was keenly aware of many specific real world applications of complex numbers well before I ever studied ComplexAnalysis, and which in fact motivated my study of the topic, and conceivably it's much more common for students to encounter ComplexAnalysis as just yet another unmotivated highly abstract topic arbitrarily required by the university, and discover applications only later. Which, if true, seems pedagogically unfortunate, but does seem to be still overall the rule, pedagogical innovations by organizations like AMS over the last decade notwithstanding. -- Doug''
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IIRC it was Rudin who said that "the most important function in mathematics" is

  e^z = sum (n=0..infinity) z^n/n! = 1 + z + z^2/2 + z^3/6 + z^4/24...
Familiarity with this and related topics in ComplexAnalysis eventually does render intuitive the identity

  e^(pi*i) = -1
''Which should obviously be written as e^(i*pi) + 1 = 0''
* Mathematicians are very, very fond of making that "correction" - quoted because it's not that the original is wrong, just that mathematicians have reason to prefer the normalized form.
** I think, Doug, that it's not the normalization that matters, but the presence of the fundamental operations. In the second form we have equality, addition, multiplication and exponentiation, and we also have 0, 1, i, e, and pi. The first form can't be viewed in that way.
** That aspect of it is pretty cool, indeed, but if that's the motivation, I disagree that it is "obviously" the right one of the two forms, considering the issues I already brought up; they both have virtues.
* But this misses the point: mathematicians aren't surprised by the identity in either form, but the point in this context is the surprise of the identity to the educated non-mathematician, and the first point makes the peculiarity very obvious: the two sides of the equation look, to the uninitiated, very much '''un'''equal, since no positive real number to any real power is negative. And even the layperson with a little familiarity with complex numbers (e.g. who know, as a factoid, that i = sqrt(-1) <=> i^2 = -1) find it peculiar that e, pi, and i somehow form this kind of interrelationship.
* Whereas the second form is not so obviously peculiar to such people; it is equating some unfamiliar-to-them formula with zero, which is extremely mundane; it's, at first glance, no different than saying f(x) = 0. "So what?"
* This is why math popularization books pretty much universally present the first form: to maximize recognition that something really different than naive intuition is at work.
* P.S. Speaking of thinking about your audience: to whom do you think it is "obvious" that your second form should be the preferred notation? The motivation that makes it "obvious" is a complete unknown to non-mathematicians, so you really need to spell out the details.

Claims by certain mathematicians to the contrary (that it can't truly be understood) strike me as either disingenuous, or possibly examples of the truism that knowledge "advances one funeral at a time".
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NovemberZeroFive

CategoryMath