existence of analytic continuation - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T22:59:57Z http://mathoverflow.net/feeds/question/69985 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69985/existence-of-analytic-continuation existence of analytic continuation Đức Anh 2011-07-11T05:07:29Z 2011-07-11T05:49:48Z <p>Good morning,</p> <p>I have just started reading Riemann surfaces. I would like to ask a question, maybe it is naive. </p> <p>Let $X$ be a Riemann surface and $\phi\in\mathcal{O}_{a,X}$ a holomorphic function germ at $a$ of $X.$ Let $u : [0,1]\to X$ be a curve, i.e a continuous mapping. Does it exist always an analytic continuation of $\phi$ along the curve $u$?</p> http://mathoverflow.net/questions/69985/existence-of-analytic-continuation/69987#69987 Answer by Robert Israel for existence of analytic continuation Robert Israel 2011-07-11T05:49:48Z 2011-07-11T05:49:48Z <p>No, e.g. you may run into a singularity. For example, take $X = {\mathbb C}$, $u(t) = t$, $a=0$ and $\phi(z) = \frac{1}{1-2z}$ in a neighbourhood of 0. The pole at $t = 1/2$ stops the analytic continuation along the curve.</p>