= Branch phase in a figure-eight analytic continuation integral
{title2=$J(z)=-2i\sin(\pi z)\int_{-1}^1e^t(1-t^2)^z\,dt$}
For a figure-eight contour around $1$ anticlockwise and $-1$ clockwise, start $(t^2-1)^z$ at $t=0$ with argument $-\pi$. Shrinking the two loops gives the displayed identity. If the interval integral instead uses $(t^2-1)^z=e^{-i\pi z}(1-t^2)^z$, the corresponding identity includes the extra factor $e^{i\pi z}$. Omitting this phase changes the value, rather than merely the locations of the meromorphic continuation's poles.
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