  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 $\gamma$(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.
