I have a question in my complex analysis worksheet.
Determine where the Cauchy Riemann conditions are satisfied for $w=\overline{z}$ (the complex conjugate of $z$).
As far as I know the function has to be differentiable for Cauchy Riemann condition to hold. But the conjugate of $z$, hasn't got a limit, so it's non differentiable. So I'm wondering how does it hold?
If assuming it is actually differentiable. $w=\overline{z}$, then $w = u+iv,$ and $z=x+iy$.
Since its a conjugate, it would be $z=x-iy$? In this case, what would $u$ and $v$ be?
If I know what $u$ and $v$ is then I can calculate the Cauchy Riemann Conditions.
Best Answer
$$w=\overline{z}$$ $$u+iv=x-iy.$$ The Cauchy-Riemann equations are:
$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$$ $$\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$$
We have $$\frac{\partial u}{\partial x} = 1 \text{ and }\frac{\partial v}{\partial y}=-1,$$
so the Cauchy-Riemann equations are satisfied nowhere.