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?

Thanks in advance.

  • 2
    $\begingroup$ It might be easier to use the Cauchy–Riemann equations: $u_x = v_y$, $u_y = -v_x$. $\endgroup$ – Martin R Feb 10 '16 at 20:58
  • $\begingroup$ Right, but I wanted to try this because the professor told that it will be useful ;) $\endgroup$ – user162343 Feb 10 '16 at 21:00

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 $v = \operatorname{Im}(f)$.

  • $\begingroup$ How do you know that the derivative of the conjugate is $0$? $\endgroup$ – user162343 Feb 10 '16 at 22:38
  • 1
    $\begingroup$ As I wrote: $\dfrac{\partial \bar f}{\partial z} = \overline{\dfrac{\partial f}{\partial \bar z}}$ which is $0$ since $f$ is holomorphic. $\endgroup$ – mrf Feb 10 '16 at 23:17
  • $\begingroup$ Aaa ok, so you are using that $\frac{\partial f}{\partial \bar{z}}=0$ only if the function is holomorphic right? $\endgroup$ – user162343 Feb 10 '16 at 23:27

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