Let $W(t\_1,\dotsc,t\_n)$ a holomorphic function on some connected open set $U$ of $\mathbb C^n$.
Let $\mathbf t^{(0)}$ a point of $U$.


Assume that there exists a cycle $\gamma$ in $\mathbb C^m$ and a rational function $F(\mathbf t, x\_1,\dotsc, x\_m)$ such that for all $\mathbf t$ in a neighbourhood of $\mathbf t^{(0)}$:


1. $\mathbf x \mapsto F(\mathbf t, \mathbf x)$ is continuous on $\gamma$;
2. $\displaystyle W(\mathbf t) = \oint\_\gamma F(\mathbf t, \mathbf x)\mathrm d \mathbf x$.


**Is it true that for all point $\mathbf t^{(1)}$ in $U$, there exists another (sum of) cycle $\gamma\_1$ such that properties 1. and 2. are satisfies in a neighbourhood of $\mathbf t^{(1)}$ ?**




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For simple integrals, that is $m=1$, this is true, and not too hard to see: the poles of $F(\mathbf t, x)$ are points that move continuously with $\mathbf t$. It is easy to deform $\gamma$ so that is does not encounter these moving points. There is a singularity when a pole inside $\gamma$ collapses with a pole outside, but this can't happen if we stay in the domain of holomorphy of $W$. However, it seems harder for multiple integrals...



Analytic continuation of a multiple contour integral
I'm not sure you're right about the $m=1$ case. Take
$$F(t,x) = \frac{2x^3}{x^2-t} = \frac{x^2}{x-\sqrt{t}} + \frac{x^2}{x+\sqrt{t}}$$
take $t^0=1$ and take $\gamma$ to be a circle in the $x$-plane which encloses $1$ but not $-1$. Then 
$$\oint\_{\gamma} F(x,t) dx = (2 \pi i) (\sqrt{t})^2 = (2 \pi i) t$$
where $\sqrt{t}$ means the square root of $t$ which lies in $\gamma$.


Now, the function $W(t) = (2 \pi i) t$ extends holomorphically to the whole $t$-plane. Consider $t^1$ to be the point $t=0$. 


If I understand correctly, you are claiming that there is a cycle $\gamma^1$ so that $F(t,x)$ will be continuous on $t \times \gamma^1$ for any $t$ near $0$, and this cycle will have $\int\_{\gamma\_1} F(t,x) dx = (2 \pi i) t$.


But $F$ is discontinuous at $(0,0)$, so $\gamma\_1$ must not pass through $0$. Thus, either $\gamma\_1$ encloses $0$ or it doesn't. So, for $t$ near but not equal to $0$, either $\gamma\_1$ encloses both poles of $F$ or neither one, and we get $\oint\_{\gamma\_1} F = 2 (2 \pi i) t$ or $0$.


In other words, I disagree with your statement "There is a singularity when a pole inside $\gamma$ collapses with a pole outside, but this can't happen if we stay in the domain of holomorphy of $W$." If I rig the pole residues carefully, I can make $W$ stay holomorphic even though the pieces it is built out of are becoming singular.


