Is it possible to derive a formula for Re(z) or Im(z) that does not use Re(z) or Im(z) in the formula? If so, what is it?
In other words, given any complex number z, where z = a+bi, and a and b are both real numbers, can we find a or b without using functions like Re(z), Im(z) or abs(z) (because abs(z) uses Re and Im in its definition)
Edit:I did not consider complex conjugation when writing this. I don't "accept" it because it is a function in terms of the individual real and imaginary components of z.
$\endgroup$ 72 Answers
$\begingroup$If you ''accept'' complex conjugation, then$$\Re(z)=\frac{z+\overline{z}}{2}$$and$$\Im(z)=\frac{z-\overline{z}}{2i}$$
$\endgroup$ 3 $\begingroup$It depends on that you mean by formula. In a wide sense, as argued below, the answer is no.
Indeed, $\Re(z)$ is not a holomorphic function since its image is the real line. In this sense, there is no formula for $\Re(z)$ that does not involve $\bar z$, because the Cauchy–Riemann equations fail for $\Re(z)$:$$ {\frac {\partial \Re(z)}{\partial {\bar {z}}}}=\frac12 \ne 0 $$
$\endgroup$ 7
