Let $G(s,t)$ be a complex valued function in two variables that converges absolutely for $Re(s), Re(t)>1$. Suppose we can analytically continue $G$ in such a way that
$$G(s,t) = f(s)g(t)H(s,t)$$
where $f(s)$ and $g(t)$ have simple poles at $s=1,t=1$, respectively, and $H(s,t)$ converges for $Re(s)>\frac{1}{k}$ and $Re(t)>\frac{1}{\ell}$ for some $k,\ell>1$. Is there a contour $\Gamma \subset \mathbb{C}^2$ such that
$$\int_\Gamma G(s,t) = Res_{s=1}(f(s))Res_{t=1}(g(t))H(1,1)$$
My first guess would be for some small $\epsilon>0$ and some $\delta>1$ to let $$\Gamma_\epsilon = \{(s,t) : Re(s)=\frac{1}{k}+\epsilon, Re(t) = \frac{1}{\ell}+\epsilon\}$$ $$\Gamma_\delta = \{(s,t) : Re(s)=\delta, Re(t) = \delta\}$$ $$\Gamma = \Gamma_\epsilon \cup \Gamma_\delta$$
However, I am not that good at multivariate complex analysis and am having problems showing whether this is correct or not.
Any solution or reference would be greatly appreciated.