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I want to solve the following:

Show that h is harmonic iff $\frac{\partial h}{\partial \overline z}$ is conjugate harmonic

My attempt:

$h$ is harmonic iff $\frac{\partial^{2} h}{\partial z \partial{ \overline{z}}}=0$ iff $\frac{\partial}{\partial z}(\frac{\partial h}{\partial \overline z})=0$

and given that

$$\frac{\partial h}{\partial z}= \frac{\overline {\partial{\overline h}}}{{\partial \overline z}}$$

then $\overline {\frac{\partial h}{\partial \overline z}}$ is analytic iff $\frac{\partial h}{\partial \overline z}$ is conjugate analytic.

Can you tell if I am right ? and If not can you help me to fix the proof? thanks a lot :)

this is what I don't know if it is true:

$$\frac{\partial h}{\partial z}= \frac{\overline {\partial{\overline h}}}{{\partial \overline z}}$$

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We have

$$2\frac{\partial h}{\partial z} = \frac{\partial h}{\partial x} - i\frac{\partial h}{\partial y},$$

and so

$$2\frac{\partial \bar{h}}{\partial \bar{z}} = \frac{\partial \bar{h}}{\partial x} + i\frac{\partial \bar{h}}{\partial y} = \overline{\frac{\partial h}{\partial x} - i \frac{\partial h}{\partial y}} = 2\overline{\frac{\partial h}{\partial z}}.$$

Dividing by $2$ and conjugating results in

$$\overline{\frac{\partial \bar{h}}{\partial \bar{z}}} = \frac{\partial h}{\partial z}.$$

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  • $\begingroup$ thanks a lot ;) then my proof is complete an correct? $\endgroup$
    – user162343
    Mar 15 '15 at 23:06
  • $\begingroup$ o no that has to be the conjugate analytic of other function :) $\endgroup$
    – user162343
    Mar 15 '15 at 23:10
  • $\begingroup$ the title is explicitly the exercise :) $\endgroup$
    – user162343
    Mar 15 '15 at 23:10
  • $\begingroup$ well if $v$ is a conjugate analytic of $u$ then $u+iv$ is analytic $\endgroup$
    – user162343
    Mar 15 '15 at 23:14
  • $\begingroup$ but in a previous exercise i have proved that h is harmonic iff $\frac{\partial h}{\partial z }$ is analytic. $\endgroup$
    – user162343
    Mar 15 '15 at 23:16

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