Complex numbers and Nonstandard Analysis

Question

A finite hyperreal number $r$ is a number defined as a sum of a real number and an infinitesimal number $\omega$: $$r=a+\omega$$ Do you know if is it possible (and useful) to define a complex number as $$c=a+i\omega\ ?$$ Thanks

Answer 1

I am not sure, whether following is helpful or related to your question:

But do you know wick rotation$[(x+t)\to (x+it)]$.. See wikipedia page.

Many systems can be changed to other systems by wick rotation. For example, take born infield equation and minimal surface equation, these are related by wick rotation of one variable.

That is if $\phi(x,t)$ is solution of Born-infeld equation then $\phi(x,it)$ will be solution to minimal surface equation.

So if we define $z= s+t\in\mathbb R$ some variable in solving Born-Infeld equation, then $z= s+it$ will be some point in domain of definition of minimal surfaces. This is one particular example. Basically $t\to it$ gives many time very new results in the transform equation.

Answer 2

Sorry, this is not an answer, but I cannot figure out how to comment on your question.

Where is this definition from? Is 0 a standard real or an infinitesimal? If the standard reals and the infinitesimals count as finite hyperreals (and they do by the half dozen definitions that I've seen), then you would need to allow both $a$ and $\omega$ to take the value 0 in order to get these numbers as sums of reals and infinitesimals.

Answer 3

The complex numbers can be defined as numbers of the form $x + i y$ where $x$ and $y$ are real. All constructions of standard analysis work in non-standard analysis, so this statement remains true in the non-standard model: hypercomplex numbers are numbers of the form $x + i y$ for hyperreal $x$ and $y$.

Whether it's useful to single out those complexes with standard real and infinitesimal imaginary part, I think it unlikely. Finite real and infinitesimal imaginary part is more likely to be useful.

For example, in complex analysis, one often constructs contours that include paths of the sort $x + i \epsilon$ to approximate the real line, or circular arcs $a + \epsilon e^{i \theta}$ to approximate a point, where $\epsilon$ is a small positive real number, and then takes the limit as $\epsilon \to 0$.

Replacing $\epsilon$ with a real infinitesimal would probably be useful.