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Question

Suppose that $v$ is a harmonic conjugate of $u$. Show that $-u$ is a harmonic conjugate of $v$.

Attempt

Using the information, it seems that I can write:
$f(x+iy) := u(x,y) + iv(x,y)$ with $f$ being harmonic
$$\implies \frac{\partial^2 u}{\partial x^2}+\frac{\partial ^2 u}{\partial y^2} = 0$$
and $f'(x+iy) = u_x + iv_x$ (since a harmonic function $u$ implies there is a holomorphic function $f$ with real part $u$).

My interpretation of the question is that I want to prove that also:
$f(x+iy) = v(x,y) + i(-u(x,y))$ with
$$\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0.$$
I'm not sure how to do this if so.

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We have that $f=u+iv$ is holomorphic and have to show that $g:=v-iu$ is holomorphic.

But this is easy, since $g=-if$.

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  • $\begingroup$ So to show $a$ is a harmonic conjugate of $b$, then I need to show there exists a holomorphic function $g$ such that $g(x+iy):= b+ia$? $\endgroup$ – Twenty-six colours Aug 2 '17 at 9:22
  • $\begingroup$ Yes. This is to show. $\endgroup$ – Fred Aug 2 '17 at 9:28
  • $\begingroup$ @Fread's sir, I just read your answer, sir you said, we have to show $g$ is holomorphic, but as you said, $g=-if$ and we know complex linear combination of analytic function are analytic, so $g$ will be holomorphic! (Is am I correct?), Further, can we also show, $u$ is harmonic conjugate of $ -v$? Since,$h= if$ is holomorphic, hence I think $u$ is harmonic conjugate of $-v$ is also true!!(am I correct?) Sir please reply... $\endgroup$ – Akash Patalwanshi Apr 19 '18 at 13:06

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