It's frequently useful in physics and engineering applications; are there any applications in theoretical economics? (If not, were there any attempts at incorporating CA that just never caught on?)
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1$\begingroup$ Maybe in econometric theory? I've only seen complex numbers when using things like characteristic functions, which can be useful in probability theory. $\endgroup$– PburgCommented Nov 19, 2014 at 4:55
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1$\begingroup$ Following @Pburg, complex numbers definitely "show up" in economics in as much as it uses mathematical tools which naturally generate complex number (such as when we linearize macroeconomic models around an equilibrium and obtain complex eigenvalues). However, I am unaware of any model or theory which "directly" rely on the properties of complex numbers as modelling tools. Maybe you could clarify your question : are you looking for the second or the first instance of use of complex analysis in economics? $\endgroup$– Martin Van der LindenCommented Nov 19, 2014 at 5:46
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1$\begingroup$ Using trivial properties of complex numbers is not complex analysis by any stretch. Otherwise pretty much all of real analysis is complex analysis---complex measures, Fourier transforms, etc. At the bare minimum, one needs to step into the world of holomorphic functions to be using complex analysis. Yes, there are some macro models where complex analysis is relevant. $\endgroup$– MichaelCommented Nov 19, 2014 at 14:04
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1$\begingroup$ Pretty clear what the OP is asking. I can provide a specific answer if on-hold is removed. $\endgroup$– MichaelCommented Nov 22, 2014 at 23:59
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1$\begingroup$ books.google.com/… An example of using complex numbers (though it's Sargent and Hansen who use mathematical tools very frequently!) So things like analyzing impulse response in frequency domain, which is used in electrical engineering but definitely also relevant in economics. $\endgroup$– John LukeCommented Nov 26, 2014 at 21:38
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4 Answers
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It should be pointed out that just because one encounters complex numbers does not mean one is doing "complex analysis", e.g. complex eigenvalues, complex Borel measures, Fourier transforms, etc. where trivial properties of complex numbers come up.
Complex analysis is a very focused subject unlike, say, real analysis, which is eclectic by comparison. At its core are holomorphic functions of one or more complex variables.
This paper
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=932693
is a specific instance of an economic model where complex analysis is used. The model solution technique used there is the identification between holomorphic functions on the unit disk and their continuation on the boundary. (The resulting function space is called the Hardy space, which contains the players' strategy spaces in the game being played in the paper.)
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Complex numbers and complex analysis do show up in Economic research. For example, many models imply some difference-equation in state variables such as capital, and solving these for stationary states can require complex analysis.
However, as others already emphasized, complex analysis is mostly a byproduct of solving equations. I'm not familiar with any paper where complex analysis is at the heart of the model.
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$\begingroup$ To add to the answer, one way to study difference equations is to use generating functions, which is where complex analysis comes in. $\endgroup$ Commented Nov 21, 2014 at 14:27
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$\begingroup$ For example, what equations in economics (outside of finance) have been solved by complex analysis. That would improve your answer, if you could list the examples you are aware of, at least in this limited sense. $\endgroup$– user218Commented Nov 23, 2014 at 13:37
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As described in the comments, you could maybe count instances in probability theory, econometrics, PDEs, or numerical analysis. But in general, besides using trivial properties of complex numbers (as @Micheal stated), the answer is no.
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Ben Tamari (1997). "Conservation and Symmetry Laws and Stabilization Programs in Economics." English.
Conservation and Symmetry Laws and Stabilization Programs in Economics Abstract: An autonomous economic system, i.e., a country, tends to be a conservative and a symmetrical system in Keynes space (Output, Money and Time [Ot, Mt ; t]), and can therefore be represented as a complex numbers system. This presentation makes it possible to aggregate (or disaggregate) the system at all levels, from the individual to the most general aggregate (and vice versa). It also offers a simultaneous solution to the problem of allocating and distributing useful resources in the market.
