1. Cauchy-Riemann

Show that h(z) = z bar (line over the z) is not analytic on any domain. Check the Cauchy Riemann equations to do this.

Similiar question on an upcoming exam and I'm not sure how to do it. Can someone please show the steps?

Thanks a lot!

2. $z=x+iy\mapsto \bar{z}=x-iy\equiv u(x,y)+iv(x,y)$. Then $u_x = 1$, $u_y=0$, $v_x = 0$, $v_y = -1$. The Cauchy Riemann equations are $u_x=v_y$ and $u_y=-v_x$. The former is never satisfied, so the function is not holomorphic on any domain.

3. Originally Posted by jzellt
Show that h(z) = z bar (line over the z) is not analytic on any domain. Check the Cauchy Riemann equations to do this.

Similiar question on an upcoming exam and I'm not sure how to do it. Can someone please show the steps?

Thanks a lot!
I'll add a little known (to some, maybe) but useful theorem:
When a function is expressed in the form $f(z, \bar{z})$, the Cauchy-Riemann relation is $\frac{\partial f}{\partial \bar{z}} = 0$.

Posting of proof available upon request.

So it's trivial to see that $f(z) = \bar{z}$ is not analytic .....

zz bar is analytic or not

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