Is $\ln|z|$ harmonic in the punctured disk

Question

1. how can i show that $\ln|z|$ is harmonic in punctured disk ?

2. also $\ln|z|$ has no harmonic conjugate in $\Bbb C\setminus\{0 \}$ but has in $\Bbb C\setminus[0, \infty)$.

Answer 1

The second part of (2) should be clear, since $\Bbb C\setminus[0,\infty)$ is simply connected. Hint for the first part:

Lemma 1 If $f$ is holomorphic in an open set $V$ then $\int_\gamma f'(z)\,dz=0$ for any smooth closed curve $\gamma$ in $V$.

Lemma 2. If $f=u+iv$ is holomorphic in some open set, where $u(z)=\ln|z|$, then $f'(z)=1/z$.

Hint for Lemma 2: The Cauchy-Riemann equations show that $$f=\frac12(u_x-iu_y).$$(You can simplfy the calculation of $u_x$ and $u_y$ by writing $u(z)=\frac12\ln(x^2+y^2)$.)

Answer 2

1. $z \mapsto \ln |z|$ is locally the real part of a determination of the complex logarithm, so it is locally harmonic, therefore harmonic on the whole punctured disc.