 Solution

Near each $a\in D$, the local branch from part (a), chosen to have value $\log |a|+i\theta (a)$ at $a$, agrees withIndeed, two logarithms of the same nonzero number differ by $2\pi i$ times an integer, and the integer is locally constant; it is zero at $a$. Hence $L$ is locally analytic and therefore analytic on $D$. This is the analytic logarithm on the positive-axis slit plane.

For $-1<\alpha <1$, an analytic branch is