True or false: Every harmonic function on $\Bbb C \setminus \{0\}$ has a harmonic conjugate.

Question

True or false:

Every harmonic function on $\Bbb C \setminus \{0\}$ has a harmonic conjugate.

I saw a question on this site:"Show Ω is simply connected if every harmonic function has a conjugate"

But I know that

1)"$\Bbb C \setminus\{0\}$ is not simply connected"

2. "If $f = u + i v$ is analytic then $u$ and $v$ are harmonic conjugate of each other"

3. "The function $u$ is harmonic if $\Delta^2u = 0$ where $\Delta$ is a Laplacian operator"

Please help me.

Answer 1

Is false. As the user Sangchul Lee said, consider $\frac{1}{2}\log|z|=\log(x^{2}+y^{2})$ in $\mathbb{C} \setminus \{0\}$. This function does not have a harmonic conjugate: see the answer of this post.