How to show that there is no analytic function $f$ on $D^*$ such that $Re(f) = log|z|$. How to show that there is no analytic function $f$ on $D^*$ such that $Re(f) = log|z|$. $D^*$ is the unit disk with zero removed. I am trying to show that if such function exists, then it must be $logz$, but $logz$ cannot be defined on $D*$, am I right?

Suppose there were such an $f.$ Letting $\log z$ denote the principal value logarithm, we then have $g(z) = f(z) - \log z$ holomorphic on $U= \{|z|<1\}\setminus (-1,0].$ Now $U$ is connected, and $\text { Re } g = 0$ on $U.$ It follows (use the open mapping theorem or the C-R equations) that $g$ is constant on $U.$ Thus $f(z) = \log z + c$ in $U$ for some constant $c.$ It follows that $f$ has the same jump discontinuity behavior near $(-1,0]$ that $\log z$ does, contradiction.
