Branch phase in a figure-eight analytic continuation integral ($J(z)=-2i\sin (\pi z){\int }_{-1}^1{e}^t(1-t^2{)}^{z}dt$)

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.