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Complex numbers are pairs of real numbers. As a vector space they are isomorphic to R^2 and one construction of the complex numbers is, in effect, just as pairs

Complex numbers are pairs of real numbers. As a vector space they are isomorphic to R^2 and one construction of the complex numbers is, in effect, just as pairs of real numbers. (There are of course other ways of constructing the complex numbers, too, such as via matrices. These are also all isomorphic to R^2 as a vector space.) Of course, what makes complex numbers special is that you can multiply them, and that gives you a field. Following that train of thought, > Are there any simple and obvious applications that require continuity/calculus or topology on the complex plane that can't be restated in terms of real-pairs (and calculus/topology over those)? You can of course develop all of complex analysis without introducing the traditional a+bi notation and just referring to vectors in R^2. That's notation. However, complex differentiability (being holomorphic) is a stronger definition than regular differentiability in C(R^2, R^2), in particular, the Cauchy-Riemann equations need to be satisfied. From that stronger definition, you get all sorts of other theorems such that all holomorphic functions are analytic and so on.

KirillPanov 5y ago
Thank you for bringing up complex differentiability. Complex analysis is the only field I've found that seems to genuinely require complex numbers rather than just polar coordinate notation. However I get the impression that complex analysis, as a research field, has been a dead end for a long time now with few people working on it [1]. If pure research has come to a halt then it's fair to judge the field based on the practical applications it has achieved. Are there many? The class of holomorphic functions is pretty rigidly constrained -- infinitely differentiable, equivalent to a Taylor series at each point. I once heard that even physicists doubt the legitimacy of assuming that all physical functions are holomorphic (but now I'm speaking beyond my expertise). Continuing to speak beyond my expertise, I don't think Feynman had holomorphic functions in mind when he said "our jewel". I have to say that the definition of limits for complex numbers always seemed like an artificial straitjacket. I understand why you can't have complex limits defined any other way, but maybe the take-away here isn't "complex limits should work like this" so much as "complex limits, and therefore complex analysis, is not very useful". [1] "Complex analysis is pretty much dead. There are some remaining problems but most of them are either too specialised or too difficult. This is not a very healthy situation, and there are relatively few people working in complex analysis nowadays.", Dmitry Beliaev , former Assistant Professor (Mathematics) at Princeton University, https://www.quora.com/Are-there-any-mathematics-faculty-at-Princeton-specializing-in-complex-analysis?share=1 https://www.quora.com/Are-there-any-mathematics-faculty-at-P...

KirillPanov 5y ago

Thank you for bringing up complex differentiability. Complex analysis is the only field I've found that seems to genuinely require complex numbers rather than just polar coordinate notation. However I get the impression that complex analysis, as a research field, has been a dead end for a long time now with few people working on it [1]. If pure research has come to a halt then it's fair to judge the field based on the practical applications it has achieved. Are there many? The class of holomorphic functions is pretty rigidly constrained -- infinitely differentiable, equivalent to a Taylor series at each point. I once heard that even physicists doubt the legitimacy of assuming that all physical functions are holomorphic (but now I'm speaking beyond my expertise). Continuing to speak beyond my expertise, I don't think Feynman had holomorphic functions in mind when he said "our jewel". I have to say that the definition of limits for complex numbers always seemed like an artificial straitjacket. I understand why you can't have complex limits defined any other way, but maybe the take-away here isn't "complex limits should work like this" so much as "complex limits, and therefore complex analysis, is not very useful". [1] "Complex analysis is pretty much dead. There are some remaining problems but most of them are either too specialised or too difficult. This is not a very healthy situation, and there are relatively few people working in complex analysis nowadays.", Dmitry Beliaev , former Assistant Professor (Mathematics) at Princeton University, https://www.quora.com/Are-there-any-mathematics-faculty-at-Princeton-specializing-in-complex-analysis?share=1 https://www.quora.com/Are-there-any-mathematics-faculty-at-P...