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I will describe my set-up and then the problem. We use the branch of the complex square root where $$ \sqrt{re^{i \phi}} = \sqrt{r} e^{i \frac{\phi}{2}} \qquad \forall r > 0 \,, \forall \phi \in [0,...

This property seems to be used both in the context of differential equations and several kinds of discrete equation systems or automata. It seems to be related in certain case to the Painlevé ...