Branches, monodromy, and Riemann surfaces

On a connected open domain U avoiding zero, a holomorphic logarithm L of z must satisfy $e^{L(z)}=z$. Differentiation gives L′=1/z. Thus every closed contour integral of 1/z in U must vanish. Conversely, if all those periods vanish, integrating 1/z from a base point defines a primitive; choosing its additive constant so that its exponential agrees with z at the base point produces a logarithm throughout U. A simply connected domain avoiding zero is sufficient because Cauchy's theorem makes the periods vanish.

The punctured plane fails this test: a positive circle has integral 2πi, so no single-valued global logarithm exists there. Removing the nonpositive real axis gives a domain with a principal branch whose argument lies between −π and π. Other cuts give other branches. Points along the chosen cut, away from zero, still have local logarithms; the cut is a choice of planar bookkeeping rather than a line of intrinsic local singularities. Zero remains a genuine obstruction to continuing a logarithm through it.

Start with a holomorphic germ, meaning a function specified near one base point with two specifications identified when they agree nearby. Continue it along a path by compatible overlapping neighborhoods. Uniqueness on overlaps follows from the identity theorem, but a closed path can lead to a different germ at the original point. This permutation or change of branches is monodromy. For a square root, one circuit around zero swaps the two choices; for a logarithm, each positive circuit adds 2πi.

A standard monodromy theorem says that a holomorphic germ admitting analytic continuation along every path in a simply connected domain determines one single-valued holomorphic function there. Both hypotheses matter: simple connectedness removes loop ambiguity, while continuation along every path excludes an unacknowledged singular obstruction. Merely drawing one path that works establishes neither a domain-wide extension nor path independence. On a multiply connected domain, continuation along homotopic fixed-endpoint paths agrees provided the germ can be continued along all paths in that homotopy.

A Riemann surface is a one-complex-dimensional manifold described by local charts with holomorphic transition maps. For the square root, pairs (z,w) satisfying w²=z form a surface parameterized by w. Away from z=0 the projection to z is a two-sheeted unbranched covering; each local inverse selects one root. Adding the point w=0 gives a regular point of the surface, while the projection ramifies there and has branch value z=0 because its derivative vanishes. The surface itself is not singular at that point.

For a logarithm, use pairs with z=eʷ. The coordinate w ranges over the complex plane, and projection to the punctured z-plane is an infinite-sheeted covering. The logarithm is simply the single-valued coordinate w on that surface. This construction does not make a global planar branch exist; it changes the domain to record continuation history. Distinguish an intrinsic surface, its projection, a selected planar branch, and a branch cut before interpreting drawings of stacked sheets.