Angular summation

Near a generic diagonal pair, $H_{F,E}$ is precisely (eq:contour) with the $k=0$ term removed. At a boundary parameter its analytic representative is the one in Theorem thm:analytic.

For $\alpha =1/N$ and $N\beta \in \mathbb{Z}$, every $\sin (\pi {\nu }_n)$ vanishes and the kernel is the full finite character-weighted quotient image kernel. For even $N$, the strict image set at zero angle omits the two boundary representatives of one quotient image. The full quotient sum or an equivalent limiting prescription must be used; simply dropping them is incorrect. For incompatible characters diffraction need not vanish.