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Given a complex number $z$ , is it possible to express its conjugate $\bar z$ in terms of $z$ using only operations of addition , subtraction , multiplication , division and exponentiation on $z$ as a whole .

In other words , expressing $z$ as $x + iy$ and then manipulating $x$ and $y$ is not allowed .

If it is not possible , can we prove that in some elementary way ?

Vivek
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No. All of those are analytic and compositions of analytic functions are analytic.

ncmathsadist
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It's impossible. The functions you're allowing are differentiable wherever they are defined. The function $\overline z$ is differentiable nowhere.