Log modulus has no global harmonic conjugate on the punctured plane
= Log modulus has no global harmonic conjugate on the punctured plane
{title2=$\log|z|$ on $\mathbb C^*$}
The function $\log|z|$ is <harmonic function>[harmonic] on the <punctured complex plane>, but it is not globally the real part of a <holomorphic function> there. Such a function would have derivative $1/z$, whose integral around the unit circle is $2\pi i$, whereas the integral of a derivative around a closed curve is zero.