Analytic continuation of a real function I know that for $U \subset _{open} \mathbb{C}$, if a function $f$ is analytic on $U$ and if $f$ can be extended to the whole complex plane, this extension is unique.
Now i am wondering if this is true for real functions. I mean, if $f: \mathbb{R} \to \mathbb{R}$, when is it true that there is an analytic $g$ whose restriction to $\mathbb{R}$ coincides with $f$ and also when is $g$ unique. 
Surely $f$ needs to be differentiable but this might not be sufficient for existance of such $g$.
edit: I mean, is it easy to see that there is and extension of sine cosine and exponential real functions?
Thanks a lot.

I think that most mathematicians would say that the functions you mention are restrictions to the real line of functions more naturally defined on the complex plane in the first place. So--yes.
