# Correct way to analytically continue a multi-dimensional integral

Consider a multi-dimensional integral

$$$$\int dx_1 \int dx_2 ... \int dx_n f(x_1,...,x_n) .$$$$

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

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

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

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