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Can the shift of center of convergence for power series from point to point in a path of overlapping circles, in analytic continuation, be interpreted as a translation in any way?

https://upload.wikimedia.org/wikipedia/commons/0/0f/Analytic_continuation_3.png

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  • $\begingroup$ As written, the question is incomprehensible. You may want to ask a friend to help you with your English or post the question in your native language and ask somebody at MSE to translate. $\endgroup$ – Moishe Kohan Apr 13 at 15:27
  • $\begingroup$ Of course, you can always use a translation to move one round disk to another round disk (of the same radius). But, in general, the radius will change, so you have to use a contraction/expansion as well (at some point). I do not think it will help you with understanding analytic continuation. My suggestion is to work out some examples such as $\sqrt{z}$ and $\log(z)$. $\endgroup$ – Moishe Kohan Apr 13 at 17:15
  • $\begingroup$ In my experience it is better to think of as analytic functions having some "inherent knowledge" of how they ought to extend locally; the problem is that because this is local, we need some additional constraint to always get the same answer when we approach in different ways (e.g. logarithm). As such, it is not quite a translation, or any geometric action for that matter. $\endgroup$ – Brevan Ellefsen Apr 13 at 23:08

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