Analytic logarithm on the positive-axis slit plane

On $\mathbb{C}\setminus [0,\infty )$, choose the argument in $(0,2\pi )$. Thenis an analytic logarithm. Locally this follows by scaling the convergent series

For $-1<\alpha <1$,with value one at $\alpha =0$ by continuity. A keyhole contour around the positive axis gives a jump factor $1-e^{2\pi i\alpha }$, while the double pole at $-1$ has residue $-\alpha e^{\pi i\alpha }$.