Contrarily to finite-dimensional systems, fractional systems have irrational transfer functions with non unique analytic continuations (from some right-half Laplace plane to a maximal domain). They involve continuous sets of singularities, namely cuts, which can be chosen arbitrarily between fixed branching points. This paper presents an academic example of the 1D heat equation and a realistic model of an acoustic pipe, both on bounded domains, which involve transfer function with a unique analytic continuation with singularities of pole type. When the length of the domain becomes infinite, these sets of singularities degenerate into uniquely defined cuts. From a mathematical point of view, both the convergence in the Hardy space of some left half-complex plane and the pointwise convergence are studied and proved.