Several complex variables and analytic spaces Definition

Analytic continuation in several variables

Builds on Holomorphic function of several variables Open set Connected set Connected component

Definition

A function element in $\C^n$ is a pair $(U, f)$ where $U \subseteq \C^n$ is a nonempty connected open set and $f$ is a holomorphic function on $U$ .

A function element $(V, g)$ is a direct analytic continuation of $(U, f)$ if $U \cap V$ is nonempty and $f = g$ on some connected component of $U \cap V$ .
A function element $(V, g)$ is an analytic continuation of $(U, f)$ along a chain if a finite sequence of function elements runs from $(U_0, f_0) = (U, f)$ to $(U_k, f_k) = (V, g)$ . Each $(U_j, f_j)$ must be a direct analytic continuation of $(U_{j-1}, f_{j-1})$ .

If $V \supseteq U$ and $g = f$ on $U$ , one says that $g$ is an analytic continuation of $f$ to $V$ .

Remarks

By the identity theorem, if $f$ and $g$ agree on a nonempty open subset of a component of $U \cap V$ , they agree on that whole component.

Example: let $U = \set{z \in \C^2 : \operatorname{Re} z_1 > 0}$ and $f(z) = \log z_1$ with the principal branch. Moving through half-spaces in the $z_1$ -direction around the plane $z_1 = 0$ gives a chain of direct continuations that returns to $U$ with the function $\log z_1 + 2\pi i$ . So continuation along a closed chain need not give back the original function element.

Non-example: if $U \cap V$ is empty, then $(V, g)$ is never a direct analytic continuation of $(U, f)$ , whatever $g$ is.

For $n = 1$ this is the usual analytic continuation of a holomorphic function along a chain of discs or domains.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Connected component concept
Connected set concept
Holomorphic function of several variables concept
Open set concept
Analytic continuation in several variables definition

Tag	`TK9WPV0EXS`
URI	`m2://TK9WPV0EXS`
Kind	definition
Writing	W0 stub
MSC	32D05, 32A10
Defines	Analytic continuation in several variables
Uses	Holomorphic function of several variables, Open set, Connected set, Connected component

Analytic continuation in several variables

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here