Symmetry of the complex plane

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

It's not always true that you can simply switch [tex]z[/tex] with [tex]\overline{z}[/tex] without consequence. For example, given an analytic function [tex]df/d\overline{z}=0[/tex]...a statement which is true of all functions! Another fact is that [tex]f(z) = \overline{z}[/tex] 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

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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.
 
a. Choosing i as the "basic imaginary unit" that satisfies [tex]x^{2}=-1[/tex] 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)
 

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