10 Analytic Continuation
The syllabus closes with this chapter and it is the one that explains a puzzle left hanging by Chapter 2. The geometric series \[\sum ^{\infty }_{n = 0} z^n\] converges only on the disc \(\left |z\right | < 1\), and there it sums to \(\frac {1}{1 - z}\). But \(\frac {1}{1 - z}\) is analytic at every point of the plane except \(z = 1\). The series is therefore not the whole of anything: it is a partial view of a function that lives on a much larger set, and the circle \(\left |z\right | = 1\) is an accident of how we chose to write it down rather than a boundary of the function itself.
The business of this chapter is to make that idea exact — to say when a function given on a small set has one and only one analytic extension to a larger one, and to explain why \(\log z\) resists having a single one.
Definition 10.1 (function element). A function element is a pair \((f, D)\) in which \(D \subseteq \mathbb {C}\) is a domain (a non-empty open connected set) and \(f\) is analytic on \(D\).
Definition 10.2 (direct analytic continuation). Two function elements \((f_1, D_1)\) and \((f_2, D_2)\) are direct analytic continuations of one another if \[D_1 \cap D_2 \neq \emptyset \quad \text {and}\quad f_1(z) = f_2(z) \quad \text {for all } z \in D_1 \cap D_2 .\] When this happens the function \[f(z) = \begin {cases} f_1(z) & z \in D_1\\ f_2(z) & z \in D_2\end {cases}\] is well defined and analytic on \(D_1 \cup D_2\), and is called the continuation of either element to that union.
Example 10.3. Let \(D_1 = \{\left |z\right | < 1\}\) with \(f_1(z) = \sum ^{\infty }_{n=0} z^n\), and let \(D_2 = \mathbb {C} \setminus \{1\}\) with \(f_2(z) = \frac {1}{1 - z}\). On \(D_1 \cap D_2 = D_1\) the two agree, so \((f_2, D_2)\) is a direct analytic continuation of \((f_1, D_1)\). The series was never the object of interest; it was one formula for it.
10.2 Continuation Along a Path
10.3 The Logarithm, and Why a Function Can Fail to be Single-Valued
10.4 Homotopy and the Monodromy Theorem
10.5 The Schwarz Reflection Principle
10.6 When Continuation is Impossible: Natural Boundaries
10.7 Practice Problems
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