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Since you ask, there has been much research on the analogous question of elements of finite order in the Nottingham group, which is the group of power series under composition: $$ N(\mathbb K) = \left\{ x + \sum_{n\ge2} c_nx : c_n\in \mathbb K\right\}. $$ especially for $\mathbb K$ a finite field. Your example $-\frac{2x}{\sqrt{x^2-4}}$ is almost of this form, it's expansion starts $ix+\text{h.o.t.}$ See this question, which is related to what you're asking about, and my answer to that question gives some references that discuss elements of finite order in the Nottingham group.