= Jump of a fractional power across a branch cut
{title2=$F_-=e^{-2\pi i\alpha}F_+$}
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 $\phi<\arg z<\phi+2\pi$. On the cut ray $z=re^{i\phi}$, the bank whose argument approaches $\phi$ has value $F_+=f(re^{i\phi})r^{-\alpha}e^{-i\alpha\phi}$, while the other bank has argument approaching $\phi+2\pi$ 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.
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