6. Lemma 2.8.1 - Analytic Continuation along a curve

Suppose we have two disk chains along a curve $γ : [t_{0}, t_{1}] \to C$ , say $D_{*} = ((D_{0}, d_{0}), \dots, (D_{n}, d_{n}))$ and $C_{*} = ((C_{0}, c_{0}), \dots, (C_{n}, c_{m}))$ (we also impose the condition that $(D_{0}, d_{0}) = (C_{0}, c_{0})$ ). This in particular means that $c_{m} = d_{n} = γ (t_{1})$ , so $C_{m}$ and $D_{n}$ are both neighborhoods of the last point. Let $g : C_{m} \to C$ be an analytic continuation of $f : C_{0} \to C$ along $C_{*}$ , and $\tilde{g} : D_{n} \to C$ be an analytic continuation along $D_{*}$ . Then 1;; $g = \tilde{g}$ on the smaller of the two disks $C_{m}$ and $D_{n}$ .

Remark

To prove this, we used a power series expansion at each point $t \in [t_{0}, t_{1}]$ such that the power series' agree locally. These local power series are a sort of invariant of our disk chain along $γ$ , and therefore it makes sense to talk about THE analytic continuation of $f : C_{0} \to C$ along $γ$ .