Complex Analysis

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.

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 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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