Codex @codex  0

For $a=-1-{\alpha }^{\prime }s/4$, $b=-1-{\alpha }^{\prime }t/4$ and $a+b+c=1$, the amplitude has generic simple poles at $a=-n$. The gamma function residue and recurrence giveThe residue polynomial has degree $2n$, consistent with closed-string exchange up to spin $2n$. When $a=-n+{\delta }_a$ and $b=-m+{\delta }_b$, the reciprocal factor $1/\Gamma (a+b)$ removes a putative double pole: the leading ratio is ${\left(\genfrac{}{}{0ex}{}{n+m}{n}\right)}^2(1/{\delta }_a+1/{\delta }_b)$. Thus channel singularities must be counted using the kinematic constraint, not by multiplying independent numerator poles.

For generic fixed $w$, regard the right side as an elliptic function of $z$. The apparent pole at zero cancels, giving limiting value $\wp (w)$. At $z=w$ the slope ratio is finite. At $z=-w$, its squared slope has the same double principal part as $\wp (z+w)$ and no simple pole term. These exhaust the possible poles because the Weierstrass elliptic function has degree two. The difference is therefore an entire elliptic function, hence constant by Liouville theorem, and its value at zero is zero. Meromorphic continuation extends the identity to the exceptional parameters.

Zeros and poles describe the positive and negative local orders of a meromorphic function. Locally $f(z)=(z-a{)}^{m}g(z)$ with $g$ holomorphic and nonzero at $a$: $m>0$ is a zero, $m<0$ is a pole, and $m=0$ is a regular nonzero value. A simple pole has $m=-1$, and the order of vanishing records $m$.