Calculate $ \oint_\gamma \frac{\ln(1 - \overline z)}{z - w} dz$ I'm trying to calculate
$$ \oint_\gamma \frac{\ln(1 - \overline z)}{z - w} dz$$
where I'm taking the principal branch of the logarithm, $\gamma$ is a smooth curve in the complex plane and $w \in \mathbb C$.
If $\gamma$ is the unit circle and $|w| \leq 1$ (that is, $w$ is in the domain inscribed by $\gamma$) I believe the answer is just $0$.  I'd like to expand the result to a larger class of curves, but the conjugate in the log term is throwing me for a loop (pun!).
Can anyone solve this for a wider class of curves?
 A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
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I'll consider the second paragraph case $\ds{\pars{~\mbox{"If}\ \gamma\ \mbox{is the unit circle and}\ w \in \gamma\ldots\mbox{"}~}}$:
\begin{align}
&\bbox[5px,#ffd]{\left.\oint_{\verts{z}\ =\ 1}
{\ln\pars{1 - \overline{z}} \over z - w}\,\dd z
\,\right\vert_{\ \verts{w}\ <\ 1}}
\\[5mm] = &\
\oint_{\verts{z}\ =\ 1}
{\ln\pars{1 - 1/z} \over z - w}\,\dd z
\\[5mm] \stackrel{z\ \mapsto\ 1/z}{=}\,\,\,&
\oint_{\verts{z}\ =\ 1}\,
{\ln\pars{1 - z} \over 1/z - w}\,{\dd z \over z^{2}}
\\[5mm] = &\
{1 \over w}\oint_{\verts{z}\ =\ 1}\,\,\,
{\ln\pars{1 - z} \over z\pars{1/w - z}}\,\dd z
\\[5mm] &\ = \bbx{\large 0} \\ & 
\end{align}
