Analytic logarithm on the positive-axis slit plane
= Analytic logarithm on the positive-axis slit plane
On $\mathbb C\setminus[0,\infty)$, choose the argument in $(0,2\pi)$. Then
$$
L(z)=\log|z|+i\arg z
$$
is an analytic logarithm. Locally this follows by scaling the convergent series
$$
-\sum_{n=1}^{\infty}\frac{(1-z)^n}{n},
\qquad |z-1|<1.
$$