= Solution
Near each $a\in D$, the local branch from part (a), chosen to have value $\log|a|+i\theta(a)$ at $a$, agrees with
$$
L(z)=\log|z|+i\theta(z).
$$
Indeed, 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
$$
\boxed{z^\alpha=\exp(\alpha L(z)).}
$$
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