$f,\overline f$ are both analytic in a domain $\Omega$ then $f$ is constant?

is it true if $f,\overline f$ are both analytic in a domain $\Omega$ then $f$ is constant? I am not able to find out what property of holomorphic map I need to apply. please help.Thank you.

$f(z)=u(x,y)+iv(x,y)$, $\bar{f}(z)=u(x,y)-iv(x,y)$

Solutions Collecting From Web of "$f,\overline f$ are both analytic in a domain $\Omega$ then $f$ is constant?"

Let $f(z) = u(x,y) + i v(x,y)$, then $\bar f(z) = u(x,y) – iv(x,y)$.

  • Cauchy-Riemann equations for $f$: $(1) \quad \partial_xu = \partial_yv$, $(2) \quad \partial_yu = -\partial_xv$
  • Cauchy-Riemann equations for $\bar f$: $(3) \quad \partial_xu = -\partial_yv$, $(4) \quad \partial_yu = \partial_xv$

Combine $(1)+(3)$ to get $\partial_xu = \partial_yv = – \partial_xu$, therefore $\partial_xu = \partial_yv = 0$. Similarly, $(2)+(4)$ imply that $\partial_yu = \partial_xv = 0$. $u$ and $v$ have all their partial derivatives vanishing, $\Omega$ is connected (being a domain), therefore $u$ and $v$ are constant, and $f$ is a constant.