What is the significance of symmetry in the complex plane?

[lolgarithms]

How does one express mathematically the fact that:
if we complex-conjugated everything (switch i to -i (j to -j etc. in hypercomplex numbers) in all the definitions, theorems, functions, variables, exercises, jokes ;-)) in the mathematical literature the statements would still be true?

 

[rsq_a]

Hmmm. Well, that's a very open-ended question.

It's not always true that you can simply switch $$z$$ with $$\overline{z}$$ without consequence. For example, given an analytic function $$df/d\overline{z}=0$$...a statement which is true of all functions! Another fact is that $$f(z) = \overline{z}$$ is a nowhere analytic map, which means that you can't simply conjugate things without worrying about consequences!

However, in the largest picture, I suppose your question is about symmetry in the complex plane. In that case, I suppose the key lies in the fact that analytic functions (and thus satisfying the Cauchy-Riemann equations) impose a restriction on how the function behaves as x and y vary...that is, the (x,y) coordinates are necessarily coupled. This results in a great deal of symmetry.

 

[Landau]

If you switch i to -i in the definition of i (i.e. in the definition of complex numbers themselves), then you're just renaming/relabelling a symbol.

 

[elibj123]

a. Choosing i as the "basic imaginary unit" that satisfies $$x^{2}=-1$$ is arbitrary, you could have chosen to work with -i instead.

b. (Equivalent) Measuring the argument counter-clockwise is also arbitrary, and you except that the sames conclusions and theorems will remain unchanged when measuring the argument clockwise (which is equivalent to conjugating)