Continuation along overlaps
Definition. An analytic continuation extends a holomorphic function from one domain to a larger connected domain so that the extensions agree on overlaps. The identity theorem makes such continuations unique whenever the overlap is connected and nonempty.
Continuation along chains
Proposition. A germ can be continued along a path by a chain of overlapping discs. At each step, the power series in the next disc is determined by agreement on the overlap.
Monodromy theorem
Theorem. If analytic continuation of a germ along paths in a simply connected domain is always possible, then the continuation is single-valued. Homotopic paths give the same branch by uniqueness and deformation through overlapping chains.
Branch phenomena
Example. The logarithm can be continued locally on \(\mathbb C\setminus\{0\}\), but continuation around a loop encircling zero changes the value by \(2\pi i\). A simply connected slit domain removes this ambiguity.
Natural boundaries
Remark. Some power series cannot be continued past any point of their circle of convergence. Analytic continuation is powerful, but singularities may form a genuine barrier.
Germs
Definition. A germ at \(a\) is an equivalence class of holomorphic functions agreeing on some neighborhood of \(a\). Analytic continuation is best understood as transporting germs along paths.
Germs and continuation chains
A germ at \(z_0\) is an equivalence class of holomorphic functions agreeing near \(z_0\). A continuation along a path is a finite or locally finite chain of discs covering the path, with holomorphic representatives agreeing on overlaps. The identity theorem makes each next germ unique once the chain is fixed.
Monodromy
Theorem (monodromy). Suppose a germ at \(z_0\in\Omega\) can be analytically continued along every path in a simply connected domain \(\Omega\), and continuation depends continuously on path deformations. Then all terminal germs at a point agree and define a single-valued holomorphic function on \(\Omega\).
Proof. Any two paths with the same endpoints are homotopic relative to endpoints. Subdivide the homotopy square finely enough that each small cell lies in a continuation disc. Moving across one cell does not change the terminal germ because representatives agree on overlaps. Induction over the grid proves equality along homotopic paths. Simple connectedness makes all endpoint paths homotopic. Local representatives then patch by the identity theorem.
Branches of logarithms and roots
Proposition. A nonvanishing holomorphic function \(f\) on a domain has a holomorphic logarithm if and only if \[ \int_\gamma\frac{f'}f\,\mathrm{d}z=0 \] for every closed contour \(\gamma\). In particular, this holds on a simply connected domain.
Proof. If \(f=e^g\), then \(f'/f=g'\) has zero periods. Conversely, zero periods make \[ g(z)=\int_{z_0}^z\frac{f'}f\,\mathrm{d}\zeta+\log f(z_0) \] well defined. Differentiating \(e^{-g}f\) gives zero, and its value at \(z_0\) is one.
An \(n\)th root exists precisely when the winding numbers of \(f\circ\gamma\) are divisible by \(n\). The multivaluedness of \(\log z\) and \(z^\alpha\) on \(\mathbb C\setminus\{0\}\) records the fundamental group.
Riemann surfaces and natural boundaries
When continuation around loops changes the germ, the continuations become single-valued on a covering Riemann surface. The logarithm lives naturally on the universal cover of \(\mathbb C^\times\); algebraic functions live on branched covers. A natural boundary is a boundary across which no germ can be continued at any point, as occurs for many lacunary power series. Radius of convergence alone does not distinguish an isolated singularity from such a dense obstruction.