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Arc length as a conformal parameter for locally analytic curves

For any locally analytic curve we show that arc length can be complexified and seen as a conformal parameter. As an application, we show that any such curve defines a unique maximal one and that the notions of analytic Jordan curve coincides with the notion of a Jordan curve which is locally analytic. We give examples where we also find, for a curve $γ$(s), the limit sets of the largest extensions, that is, the limit set of the curve as s converges to the end point of the interval.