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On a simply connected domain, pathwise continuation glues to one holomorphic function

Statement

Let $Ω \subseteq C$ be simply connected, let $a_{0} \in Ω$ , and let $ξ_{0}$ be a holomorphic germ at $a_{0}$ that admits analytic continuation along every path in $Ω$ starting at $a_{0}$ . Then there is a holomorphic function $F : Ω \to C$ such that for every path $γ$ in $Ω$ starting at $a_{0}$ , the terminal germ of the continuation of $ξ_{0}$ along $γ$ is exactly the germ of $F$ at $γ (1)$ .

Facts & Assumptions

Given: A simply connected complex domain $Ω$ , a base point $a_{0} \in Ω$ , and a germ $ξ_{0}$ at $a_{0}$ that admits continuation along every path from $a_{0}$ .

[L1]

Fixed-endpoint path-homotopic paths give the same terminal germ (Fixed-endpoint homotopic paths give the same analytic continuation).

[L2]

A simply connected space is nonempty, path-connected, and has trivial fundamental group at every basepoint (Simply connected topological spaces).

[L3]

A based loop class is the class of a loop modulo endpoint-fixed path homotopy, and the constant loop is the identity element (Based loops and the fundamental group, Loop classes form the group $π_{1} (X, x_{0})$ under concatenation).

Proof

technique · direct

1.1L2choose

Fix $z \in Ω$ . Because $Ω$ is path-connected by [L2], there is at least one path $γ_{z}$ from $a_{0}$ to $z$ . Let $T_{γ}$ denote the terminal germ obtained by continuing $ξ_{0}$ along a path $γ$ from $a_{0}$ .

1.2L2L3

If $γ$ and $β$ are two paths from $a_{0}$ to $z$ , then $λ : = \overset{ˉ}{γ} * β$ is a based loop at $z$ . By [L2] and [L3], its loop class is the identity, so there is an endpoint-fixed path homotopy $K : [0, 1] \times [0, 1] \to Ω$ from $λ$ to the constant loop $c_{z}$ .

2.1step 1.2algebra

Define $e : [0, 1] \to [0, 1]^{2}$ by $e (t) = (0, 3 t)$ for $0 \leq t \leq \frac{1}{3}$ , $e (t) = (3 t - 1, 1)$ for $\frac{1}{3} \leq t \leq \frac{2}{3}$ , and $e (t) = (1, 3 - 3 t)$ for $\frac{2}{3} \leq t \leq 1$ . For $(s, t) \in [0, 1]^{2}$ , put $q_{t} (s) : = (1 - s) (\frac{1}{2}, 0) + s e (t)$ and $H (s, t) : = K (q_{t} (s))$ . Then $H$ is continuous, $H (0, t) = K (\frac{1}{2}, 0) = λ (\frac{1}{2}) = a_{0}$ , and $H (1, t) = K (e (t)) = z$ because $e (t)$ lies on the three edges where $K$ is constantly $z$ . Also $q_{0} (s) = ((1 - s) / 2, 0)$ and $q_{1} (s) = ((1 + s) / 2, 0)$ , so $H (s, 0) = λ ((1 - s) / 2) = γ (s)$ and $H (s, 1) = λ ((1 + s) / 2) = β (s)$ . Thus $H$ is a path homotopy from $γ$ to $β$ relative to the endpoints.

3.1L1step 2.1

Fact [L1] applied to the path homotopy of step 2.1 gives $T_{γ} = T_{β}$ . Therefore the value of the terminal germ over $z$ is independent of the chosen path from $a_{0}$ to $z$ .

4.1step 3.1

Define $F (z)$ to be the value at $z$ of this common germ. This is well defined by step 3.1.

5.1step 3.1step 4.1given∎

Let $z \in Ω$ and choose a path $γ$ from $a_{0}$ to $z$ . Let $(f, U)$ represent the terminal germ $T_{γ}$ at $z$ . For every $w \in U$ , the same function element $(f, U)$ continues that germ from $z$ to $w$ , so step 3.1 forces the terminal germ over $w$ to be $[f]_{w}$ . Hence $F ∣_{U} = f$ , and $F$ is holomorphic on $U$ . Since $z$ was arbitrary, $F$ is holomorphic on all of $Ω$ , and its germ at each point is the continued germ.

Depends on

Used by

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Lars V. Ahlfors, Complex Analysis, 3rd ed., Ch. 8 §1.6 (standard reference, not scraped)
Henry Wilton, Riemann Surfaces lecture notes, Corollary 9.5 (standard reference, not scraped)