Meaning of $f(z,\bar{z})$ What is the meaning of having $\bar{z}$ in your description of a function, i.e. $f(z,\bar{z})$? The conjugate is simply a function of the complex number, z, so I don't understand why you need to reference it as a second variable. It seems equivalent to noting $x^2$ as a variable in some equation like $f(x,x^2)=x+x^2$. Why does this happen?

I don't think this was a good answer, but other peoples' comments are better than the answer so I'm not deleting it.
The function $x + x^2$ is easily expressed as a function of $x$, but how would you express $z + \bar z^2$ as a function of z ?
You could do it using functions $Re$ and $Im$ which project the real and imaginary parts and get $z + Re(z)^2 + Im(z)^2 -2iRe(z)Im(z)$, but it's clumsy isn't it.
