Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / 4 / a / Solution

Specify the branch before defining a signed jump. Choose $\varphi <\arg z<\varphi +2\pi$, and let $F_{+}$ be the value on the side with argument tending to $\varphi$ from above, while $F_{-}$ is the value on the other side, with argument tending to $\varphi +2\pi$ from below. The entire function $f$ has the same limiting value on both sides, soThus the jump of a fractional power across a branch cut isEquivalently 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.