Analytic continuation of a multiple contour integral 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 $\mathbf t^{(0)}$:


*

*$\mathbf x \mapsto F(\mathbf t, \mathbf x)$ is continuous on $\gamma$;

*$\displaystyle W(\mathbf t) = \oint_\gamma F(\mathbf t, \mathbf x)\mathrm d \mathbf x$.


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 $\mathbf t^{(1)}$ ?

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...
 A: 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.
