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Translation invariance of Lauricella integral

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Given -$$f(z_0,\cdots,z_n)=\int_{\Gamma} \frac{d \zeta}{(z-z_0)^{\mu_0}\cdots(z-z_n)^{\mu_n}}$$where $\Gamma$ is an appropriate contour such that it has no poles on it. I need to show that -$$f(z_0+a,\cdots, z_n+a)= f(z_0,\cdots, z_n)$$for a small $a \in \mathbb{C}$ (Looijenga 2005). My approach to this is - take $z_i+a-\zeta=w_i$, which captures the translation. My question is, where does the condition small $a$ come into place in showing that the two integrals are equal?


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