[Math] Harmonic Conjugates — proving if $v$ is a harmonic conjugate of $u$, then $-u$ is harmonic conjugate of $v$

Question

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.

Answer 1

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

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