<p>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. </p>

<p>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. </p>

<p>This paper</p>

<p><a href="http://papers.ssrn.com/sol3/papers.cfm?abstract_id=932693">http://papers.ssrn.com/sol3/papers.cfm?abstract_id=932693</a></p>

<p>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 <em>Hardy space</em>, which contains the players' strategy spaces in the game being played in the paper.)  </p>

CC BY-SA 3.0
2014-11-28T07:42:14.960Z
2014-11-28T07:57:43.653Z
2014-11-28T07:57:43.653Z
Michael
<p>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.</p>

<p>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.</p>

CC BY-SA 3.0
2014-11-21T14:27:23.020Z
To add to the answer, one way to study difference equations is to use generating functions, which is where complex analysis comes in.
Jyotirmoy Bhattacharya
CC BY-SA 3.0
2014-11-19T17:15:27.093Z
2014-11-19T17:15:27.093Z
FooBar
<p>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.</p>

CC BY-SA 3.0
2014-11-19T17:11:06.363Z
2014-11-19T17:11:06.363Z
jmbejara
<p><img src="https://i.sstatic.net/V6FwQ.png" alt="enter image description here">Ben Tamari (1997). "Conservation and Symmetry Laws and Stabilization Programs in Economics." English.</p>

<p>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 <strong>complex numbers system</strong>. 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.</p>

CC BY-SA 3.0
2015-07-02T06:56:21.500Z
2015-07-02T06:56:21.500Z
user5087
<p>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?)</p>

<p>See <a href="http://en.wikipedia.org/wiki/Complex_analysis">http://en.wikipedia.org/wiki/Complex_analysis</a>.</p>

CC BY-SA 3.0
2014-11-26T21:38:27.030Z
http://books.google.com/books?id=hg7EQdzI_TQC&pg=PA1136&lpg=PA1136#v=onepage&q&f=false 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.
John Luke
CC BY-SA 3.0
2014-11-28T19:49:55.033Z
I think the main use of complex analysis in economics is by grad school applicants, who use good good grades in analysis courses to signal their mathematical aptitude to admissions committees.
Ubiquitous
CC BY-SA 3.0
2015-07-04T00:34:50.697Z
This comes probably closest to being a genuine example: http://www.nbs.rs/export/sites/default/internet/latinica/90/90_9/AlexeiBulatov_wp.pdf
Michael Greinecker
CC BY-SA 3.0
2014-11-19T04:55:39.933Z
Maybe in econometric theory? I've only seen complex numbers when using things like characteristic functions, which can be useful in probability theory.
Pburg
CC BY-SA 3.0
2014-11-19T05:46:58.703Z
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?
Martin Van der Linden
CC BY-SA 3.0
2014-11-19T10:30:13.270Z
Maybe in models which use differential equation. Once the eigenvalues are found, they determine dynamics like spirals.
Lex
CC BY-SA 3.0
2014-11-19T14:04:41.347Z
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.
Michael
CC BY-SA 3.0
2014-11-20T16:20:43.007Z
This question needs some clarification. Does the OP have specific examples of complexity used in physics and engineering applications? Just to get an idea of what is being asked. OP isn't asking about complexity analysis from computer science, right? (As in, complexity of algorithms?)
CompEcon
CC BY-SA 3.0
2014-11-21T02:30:16.707Z
I think the OP was pretty clear in what he/she was asking; this just happens to be a somewhat "weak" question that could be answered with a simple Google search. One vote from me to repoen.
daOnlyBG
CC BY-SA 3.0
2014-11-22T23:59:47.277Z
Pretty clear what the OP is asking. I can provide a specific answer if on-hold is removed.
Michael
<mathematical-economics>
Is complex analysis used in economics?
sirallen
# Is complex analysis used in economics?

(Asked by: username_0 on 2014-11-19)

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?)

See <http://en.wikipedia.org/wiki/Complex_analysis>.

**Tags:** mathematical-economics
> **Comments:**
> **username_4**: Maybe in econometric theory? I've only seen complex numbers when using things like characteristic functions, which can be useful in probability theory.  
> **username_5**: 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?  
> **username_6**: Maybe in models which use differential equation. Once the eigenvalues are found, they determine dynamics like spirals.  
> **username_7**: 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.  
> **username_8**: This question needs some clarification. Does the OP have specific examples of complexity used in physics and engineering applications? Just to get an idea of what is being asked. OP isn't asking about complexity analysis from computer science, right? (As in, complexity of algorithms?)  
> **username_9**: I think the OP was pretty clear in what he/she was asking; this just happens to be a somewhat "weak" question that could be answered with a simple Google search. One vote from me to repoen.  
> **username_7**: Pretty clear what the OP is asking. I can provide a specific answer if on-hold is removed.  
> **username_1**: http://books.google.com/books?id=hg7EQdzI_TQC&pg=PA1136&lpg=PA1136#v=onepage&q&f=false 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.  
> **username_2**: I think the main use of complex analysis in economics is by grad school applicants, who use good good grades in analysis courses to signal their mathematical aptitude to admissions committees.  
> **username_3**: This comes probably closest to being a genuine example: http://www.nbs.rs/export/sites/default/internet/latinica/90/90_9/AlexeiBulatov_wp.pdf  

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## 4 answers

### Answer by username_7 on 2014-11-28. Score: 18

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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### Answer by username_10 on 2014-11-19. Score: 9

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.

> **Comments:**
> **username_11**: To add to the answer, one way to study difference equations is to use generating functions, which is where complex analysis comes in.  

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### Answer by username_12 on 2014-11-19. Score: 2

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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### Answer by username_13 on 2015-07-02. Score: 1

![enter image description here](https://i.sstatic.net/V6FwQ.png)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.


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