I am trying to develop a theory explaining analytic continuation of a holomorphic function $f(z)$ defined on an open set $D \subset \mathbb{C}$. Recently, I was looking at the last chapter of Lars Ahlfors Complex Analysis book and I discovered striking similarities between my approach and that of Weierstrass.


First, lets start with a definition of holomorphy. Lets say that a complex valued function $f(z)$ is holomorphic at $z\_0$ if $\lim\_{z \to z\_0}\frac{f(z)-f(z\_0)}{z-z\_0}$ exists and is finite. So holomorphy is implicitly a local property that not only involves a single point $z\_0$ but all of the points in a neighborhood containing $z\_0$.


Now let us proceed and try to develop a global theory of holomorphic functions. The idea is to fix a point $z\_0$ in the complex plane and look at the behavior of a particular holomorphic function $f(z)$ in a neighbored $B(z\_0, r)$. It will become more clear why I take this approach. Now we want to extend the domain of holomorphy of this fixed function $f(z)$ defined initially in an open neighborhood of $z\_0$. One way to do this is to use an equivalence relation.


We say that $f(z)$ R $g(z)$ if $f(z) = g(z)$ for all $z$ in any open set containing $z\_0$. This is clearly an equivalence relation which partitions the set of holomorphic extensions of $f(z)$ near $z\_0$. By choosing a representative $g$ of each equivalence class we see that there exists open sets $U\_g$ (the domain of $g$) such that $f(z) = g(z)$ on $B(z\_0, r) \cap U\_g$ and $g$ extends $f$ to $B(z\_0, r) \cup U\_g$. Now let $U\_{z\_0} = \cup\_g U\_g$ where the union is taken over all equivalence classes described above. Then $U\_{z\_0}$ is the largest open set where we can find a holomorphic extension of $f(z)$ at $z\_0$. That is, there exists a global extension $G(z)$ of $f(z)$ at $z\_0$ in the sense that $G$ is holomorphic and the domain of $G$ is $U\_{z\_0}$.


Questions and Comments:


1. Is this formulation of analytic continuation correct and is it any different from Weierstrass' approach?
2. I cannot prove existence and holomorphy of $G$ but I suspect that follows from a simple relation between the representatives of each equivalence class and $G$.
3. How to I pass from a fixed point $z\_0$ to the entire domain of $f$? If $D$ is a countable union of open sets then I think its possible.



Global theory of holomorphic functions
In general, the holomorphic extension to $U\_{z\_0}$ that you envision does not exist. Take, for example, the principal branch of the logarithm on the disk $B(1,1)$. It has a holomorphic extension to $\mathbb{C}\setminus i[0,\infty)$, and also a holomorphic extension to $\mathbb{C}\setminus -i[0,\infty)$. Yet, it has no holomorphic extension to $\mathbb{C}\setminus\{0\}$, which is the union of the mentioned two domains.


