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Let $f$ be a harmonic function. Prove that $\overline{f}$ is harmonic.

I need help to write a rigorous proof. Thank you

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What definition are you using for "harmonic"? Can you show that $\overline{f}$ satisfies those conditions if $f$ does? –  Christopher A. Wong Sep 21 '12 at 1:19
    
f(x,y)= u(x,y) + iv(x,y) if f is harmonic then $f_{xx}$ +$f_{yy}$=0 –  yagmur Sep 21 '12 at 1:29
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OK, so try calculating $\overline{f}_{xx} + \overline{f}_{yy}$, since you know that $\overline{f}(x,y) = u(x,y) - iv(x,y)$. –  Christopher A. Wong Sep 21 '12 at 1:30
    
@yagmour: that is not the definition... It is exactly backwards: if $f_{xx}+f_{yy}=0$, then $f$ is (by definition) harmonic. –  Mariano Suárez-Alvarez Sep 21 '12 at 18:26

1 Answer 1

up vote 1 down vote accepted

We have $f_{xx}=\partial_{xx} u(x,y)+i\partial_{xx}v(x,y)$ and the same for the derivative in $y$, hence $\Delta f(x,y)=\Delta u(x,y)+i\Delta v(x,y)$. As $u$ and $v$ admit real values, so does $\Delta u$ and $\Delta v$, hence $\Delta u=0=\Delta v$.

As $\Delta\bar f(x,y)=\Delta u(x,y)-i\Delta v(x,y)$, we get that $\bar f$ is harmonic if (and only if) so is $f$.

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