I have the following complex function that I want to compute its poles and corresponding residues \begin{align} \hat{P}(u_1,u_2)=\frac{\Gamma(i(u_1+u_2)-1) \Gamma(-iu_2)}{\Gamma(iu_1+1)}, \end{align} where $\Gamma$ is the complex gamma function defined for $\Re(u)>0$.
I know that the poles of the Gamma function are located at $\{0,-1,-2,\dots\}$ with corresponding residues $\text{Res}_{-n}=\frac{(-1)^n}{n!}$. I find some confusion to determine the poles and corresponding residues for this two dimensional complex function. Analyzing each term aside, I obtain
$\Gamma(i(u_1+u_2)-1)$ has poles for $u_1+u_2=-i,0,\: i,\dots,n \:i,\: n \in \mathbf{N} $
$\Gamma(-iu_2)$ has poles for $u_2=0,\: -i,\dots,-n \:i,\: n \in \mathbf{N} $
$\Gamma(iu_1+1)$ has poles for $u_1= i,\dots,n \:i,\: n \in \mathbf{N} $
After removing the zeros from the poles, I end up with the following poles (if I did not do any mistake): $(u_1,u_2) \in \{u_1= k i,u_2= -(k+1)i, \dots,0 \}_{k \in \mathbf{N} \setminus\{0 \}} \}$ with corresponding residues $\{\frac{1}{(k+1) k}\}_{k \in N \setminus\{0\}}.$
I wanted to check if my computation is correct? is there a systematic way (references) to compute poles as well as residues for high dimensional complex valued functions? Thank you.
