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$u,v$ are harmonic conjugate with each other in some domain , then we need to show

$u,v$ must be constant.

as $v$ is harmonic conjugate of $u$ so $f=u+iv$ is analytic.

as $u$ is harmonic conjugate of $v$ so $g=v+iu$ is analytic.

$f-ig=2u$ and $f+ig=2iv$ are analytic, but from here how to conclude that $u,v$ are constant? well I know they are real valued function, so by open mapping theorem they are constant?

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  • $\begingroup$ Yes. You are completely correct. $\endgroup$ – Potato Jun 21 '13 at 7:31
  • $\begingroup$ @Wow sir, how you say by open mapping theorem they are constant? I used different method to say they are constant: let say $h= 2u$ and $h$ is analytic, so $h$ is analytic function whose imaginary part is constant hence $h$ is constant. So $u$ is constant. Similarly, say $k=2iv$ which is analytic and hence $k$ is analytic function whose real part is constant and hence $k$ is constant. So $v$ is constant. Is this method is fine? and after concluding that $u$ and $v$ are constant , can we say $f$ is constant ? sir Please reply. $\endgroup$ – Akash Patalwanshi Apr 19 '18 at 2:54
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    $\begingroup$ You are alos correct Akash $\endgroup$ – Marso Apr 19 '18 at 5:13
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Your proof is correct. I add some remarks:

  1. $v$ is a conjugate of $u$ if and only if $-u$ is a conjugate of $v$ (since $u+iv$ and $v-iu$ are constant multiples of each other)
  2. Since the harmonic conjugate is unique up to additive constant, the assumption that $u$ is a conjugate of $v$ implies (because of 1) that $u=-u+\text{const}$, and conclusion follows.
  3. Related to 1: the Hilbert transform $H$ satisfies $H\circ H=-\text{id}$.
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