Premise: I'm an undergraduate student, so I'm only considering the math perspective of the following problem and not its physical relevance nor significance.
I'm currently studying Euler's Gamma function $\Gamma(z)$, and I came across one of its applications to string theory in some notes. I am having trouble in understanding a statement, which looks wrong to me.
Consider Veneziano Amplitude function, defined as:
$$A_V(s,t) := \frac{\Gamma(-1-\alpha's)\Gamma(-1-\alpha't)}{\Gamma(-2-\alpha's-\alpha't)}$$
where $\alpha'$ is a (real) parameter and $\Gamma$ function is defined in the domain:
$$D:=\{z \in \mathbb{C}: z \neq 0,-1,-2,...\}$$
I already know that $Res_{z=-n} \Gamma(z) = \frac{(-1)^n}{n!}$ for $n \in \mathbb{Z}_{\le0}$.
My purpose is to calculate $Res_{\alpha's = n}\Gamma(-1-\alpha's)$ for $n \in \mathbb{Z}_{\ge -1}$. My notes do as follows:
$$Res_{\alpha's = n}\Gamma(-1-\alpha's) = Res_{w = -1-n}\Gamma(w) = \frac{(-1)^{n+1}}{(n+1)!}$$
while I consider the correct answer to be (let me call for simplicity $g(\alpha's):= -1-\alpha's$):
$$Res_{\alpha's = n}\Gamma(-1-\alpha's) = \frac{Res_{w = -1-n}\Gamma(w)}{\frac{dg}{d(\alpha's)}|_{\alpha's = n}}= -\frac{(-1)^{n+1}}{(n+1)!}$$
giving a minus sign not present in the notes. My result comes from a simple theorem on the residual of composite function in the case of simple poles (the only kind of singularities which occur in $\Gamma$ function), which I will not describe because already discussed in the previous question at the link Residue of composite functions.
What am I missing? Please let me know whether my answer is incomplete or unclear and I will edit it to a better version. Thank you in advance.
