We consider a family of boundary integral operators supported on a collection of parametrically defined bounded Lipschitz boundaries. Thus, the boundary integral operators themselves also depend on the parametric variables, leading to a parameter-to-operator map. One of the main results of this article is to establish the analytic or holomorphic dependence of said boundary integral operators upon the parametric variables, i.e., of the parameter-to-operator map. As a direct consequence we also establish holomorphic dependence of solutions to boundary integral equations, i.e., holomorphy of the parameter-to-solution map. To this end, we construct a holomorphic extension to complex-valued boundary deformations and investigate the complex Fr\'echet differentiability of boundary integral operators with respect to each parametric variable. The established parametric holomorphy results have been identified as a key property to derive best $N$-term approximation rates to overcome the so-called curse of dimensionality in the approximation of parametric maps with distributed, high-dimensional inputs. To demonstrate the applicability of the derived results, we consider as a concrete example the sound-soft Helmholtz acoustic scattering problem and its frequency-robust boundary integral formulations. For this particular application, we explore the consequences of our results in reduced order modelling, Bayesian shape inversion, and the construction of efficient surrogates using artificial neural networks.

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