Assume that we have two meromorphic functions $f(z,w)$ and $g(z,w)$, where $z$ and $w$ are complex (we are interested only in behavior on compact sets). Fix $z$ and assume that the sets of poles of $f(z,\cdot)$ and $g(z,\cdot)$ do not intersect. Then we can choose a contour $\gamma$ such that all the poles of $f(z,\cdot)$, but none of the poles of $g(z,\cdot)$, lie inside this contour, and calculate the integral $I(z)=\int_\gamma f(z,w)g(z,w)dw$.

Now assume that the poles of $f(z,\cdot)$ and $g(z,\cdot)$ coincide for certain $z$. Then (at least under certain assumptions) the function $I(z)$ is meromorphic, say, by Weierstrass Preparation. The question is: does this construction come up anywhere in the classical complex analysis, and if so, where can one read about it?

Update 1: One case where this construction comes up is when you try to construct the inverse $$ (A(z)\otimes 1+1\otimes B(z))^{-1} $$ from the inverses $f(z,w)=(A(z)-w)^{-1}$ and $g(z,w)=(B(z)+w)^{-1}$. Here $A(z)$ and $B(z)$ are holomorphic families of linear operators on two finite-dimensional spaces $V$ and $W$ and $A\otimes 1+1\otimes B$ acts on $V\otimes W$.

Update 2: what I could come up with so far. Assume that at $z=0$, both $f(z,\cdot)$ and $g(z,\cdot)$ have exactly one same pole at $w=0$. Let $P(z,w)$ and $Q(z,w)$ be the polynomials obtained from $1/f$ and $1/g$ near $(0,0)$ via Weierstrass Preparation Theorem. (They are polynomials in $w$ with coefficients holomorphic in $z$.). Using Euclid's algorithm, we find two polynomials $p(w)$ and $q(w)$ such that $Pp+Qq$ is identically zero in $w$ at $z=0$. With luck, the coefficients of $Pp+Qq$ are not identically zero in $z$; then we can use that $(Pp+Qq)/(PQ)$ is easy to integrate over the contour described above to get a meromorphic expansion for $I(z)$ at $z=0$.