Consider a multi-dimensional integral

\begin{equation} \int dx_1 \int dx_2 ... \int dx_n f(x_1,...,x_n) . \end{equation}

where $f$ has simple poles in each of the variables $x_1,...,x_n$. Is it correct to iteratively analytically continue variable by variable and apply residue theorem, meaning that I analytically continue $x_i\rightarrow z_i$, perform contour integration on

\begin{equation}\gamma_R=\{z_i\in\mathbb{C}| |z_i|=R, \ \mathcal{I}\text{m}[z_i]>0\}\cup[-R,R] \text{ with } R\rightarrow \infty\end{equation}

in complex space and then proceed to analytically continue $x_{i-1}\rightarrow z_{i-1}$, do the same and so on? If it is, does it produce the same result as analytically continuing all variables at once and using the definition of multi-dimensional residue and integrate on the contour

\begin{equation}\gamma_R^n=\{(z_1,...,z_n)| |z_i|=R, \ \mathcal{I}\text{m}[z_i]>0 \ \forall i=1,...,n \}\cup[-R,R]^n \text{ with } R\rightarrow \infty ?\end{equation}


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