I was considering the idea of analytic continuation:
Let $ U \subseteq \mathbb{R}$ be a subinterval (perhaps all of R). Then suppose we have some smooth function defined $f: U \rightarrow \mathbb{R}$ and we want to "analytically continue" this function to the complex plane.
We could look at the power series of the $f$ and set up sheafs and start expanding to find an analytic continuation OR...
We could ask "what is a holomorphic function equal to this function on the interval $U$?"
I.E. we want find $u(x,y), v(x,y)$ satisfying
$$ \frac{ \partial u}{\partial x} = \frac{\partial v}{\partial y} \\ \frac{ \partial u}{\partial y} = - \frac{\partial v}{\partial x} \\ u(x,0) = f(x), x \in U $$
Would there be a unique such $u,v$ pair? If there is such a unique pair would it then be THE "analytic continuation" of $f$ to the complex plane?