# How to conclude this proof that real and imaginary parts of holomorphic functions are harmonic.

I want to prove that if $f$ is holomorphic on an open set $\Omega$, then both the real and imaginary parts are harmonic, so I have proved that:

$$4\frac{\partial}{\partial z} \frac{\partial}{\partial \bar{z}}=4\frac{\partial}{\partial \bar{z}} \frac{\partial}{\partial z}=\Delta$$

and I know that if $f$ is holomorphic then $\frac{\partial f}{\partial \bar{z}} =0$ but I can't conclude using this facts that the real and imaginary parts are harmonic. Because I know that $\Delta f=0$, but I think that is not enough.

Can someone help me with this issue please?

• It might be easier to use the Cauchy–Riemann equations: $u_x = v_y$, $u_y = -v_x$. – Martin R Feb 10 '16 at 20:58
If you really want to do this using Wirtinger derivatives, $$u = \operatorname{Re}(f) = \frac12(f + \bar f),$$ so $$\Delta u = 2 \frac{\partial^2 f}{\partial z \partial \bar z} + 2 \frac{\partial^2 \bar{f}}{\partial z \partial \bar z} = 0$$ since $\dfrac{\partial f}{\partial \bar z} = 0$ and $\dfrac{\partial \bar f}{\partial z} = \overline{\dfrac{\partial f}{\partial \bar z}} = 0$, and similarly for $v = \operatorname{Im}(f)$.
• How do you know that the derivative of the conjugate is $0$? – user162343 Feb 10 '16 at 22:38
• As I wrote: $\dfrac{\partial \bar f}{\partial z} = \overline{\dfrac{\partial f}{\partial \bar z}}$ which is $0$ since $f$ is holomorphic. – mrf Feb 10 '16 at 23:17
• Aaa ok, so you are using that $\frac{\partial f}{\partial \bar{z}}=0$ only if the function is holomorphic right? – user162343 Feb 10 '16 at 23:27