= Solution
Specify the branch before defining a signed jump. Choose $\phi<\arg z<\phi+2\pi$, and let $F_+$ be the value on the side with argument tending to $\phi$ from above, while $F_-$ is the value on the other side, with argument tending to $\phi+2\pi$ from below. The entire function $f$ has the same limiting value on both sides, so
$$
F_+(r)=f(re^{i\phi})r^{-\alpha}e^{-i\alpha\phi},\qquad
F_-(r)=e^{-2\pi i\alpha}F_+(r).
$$
Thus the <jump of a fractional power across a branch cut> is
$$
\boxed{F_+-F_-=(1-e^{-2\pi i\alpha})f(re^{i\phi})r^{-\alpha}e^{-i\alpha\phi}.}
$$
Equivalently the ratio of the limiting values is $F_-/F_+=e^{-2\pi i\alpha}$ wherever $f$ is nonzero. Reversing the definition of the jump reverses its sign. Choosing a different branch multiplies both limits by the corresponding common phase.
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