MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Good morning,

I have just started reading Riemann surfaces. I would like to ask a question, maybe it is naive.

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 $u$?

share|cite|improve this question
up vote 6 down vote accepted

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.

share|cite|improve this answer
Thank you very much. My question is really naive. – Đức Anh Jul 11 '11 at 6:01

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.