Jump of a fractional power across a branch cut ($F_{-}=e^{-2\pi i\alpha }{F}_{+}$)

For an entire function $f$, let $F(z)=f(z)z^{-\alpha }$ with noninteger real $\alpha$, and choose the branch of the complex logarithm with $\varphi <\arg z<\varphi +2\pi$. On the cut ray $z=r{e}^{i\varphi }$, the bank whose argument approaches $\varphi$ has value $F_{+}=f(r{e}^{i\varphi })r^{-\alpha }{e}^{-i\alpha \varphi }$, while the other bank has argument approaching $\varphi +2\pi$ and value $F_{-}=e^{-2\pi i\alpha }{F}_{+}$. The signed jump is therefore $(1-e^{-2\pi i\alpha })F_{+}$ when defined as $F_{+}-F_{-}$. Stating the branch and order of subtraction removes the sign ambiguity. A keyhole contour uses this phase relation to convert its two bank integrals into a constant multiple of one real or radial integral.