= Index of a Wiener-Hopf kernel
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{title2=$\operatorname{ind}F=\frac1{2\pi i}\int_{-\infty}^{\infty}\frac{F'(x)}{F(x)}dx$}
For a nonvanishing kernel tending to the same nonzero value at both ends, the index counts its winding as the contour is traversed from left to right. Index zero permits a continuous logarithm tending to the same normalization at both ends. Canonical bounded <Wiener-Hopf factorization> and a constant common entire remainder require this qualification. For example $F(z)=(z+i)/(z-i)$ has index $-1$ on the real line; the bounded half-plane functions $w_+=(z-i)/(z+i)$ and $w_-=1$ obey $Fw_+=w_-$. The analytic factors $F_+=z+i$, $F_-=z-i$ make their common product $z-i$, demonstrating why bounded input functions alone do not guarantee a constant factored remainder.
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