a Riemann surface
/ REE-mahn /
Multivalued functions are awkward: over a single point z the logarithm has infinitely many values, and the square root has two, so 'the graph' over the plane is a tangled, overlapping mess. Riemann's beautiful idea was to fix the domain, not the function. Instead of forcing a multivalued function onto the flat plane, build a new, curved surface — laid out above the plane in several layers — designed so that each value of the function sits over its base point at its own distinct height. On this larger surface the function becomes perfectly single-valued: as you wander over the surface, the function never has to choose between competing values because each value lives at its own location.
Formally, a Riemann surface is a connected surface that locally looks like a piece of the complex plane, with the patches glued together by holomorphic transition maps so that 'holomorphic' makes sense globally. It is exactly a one-dimensional complex manifold. The everyday way to build the one for a given multivalued function is the gluing construction: take several copies of the plane (or of slit planes), one for each branch — these are the sheets — and glue them to each other along the cuts so that walking off the edge of one sheet brings you smoothly onto the next, in just the way analytic continuation around a branch point shuttles you from one branch to another. The points where the sheets join are the branch points.
Why this is one of the great ideas of the subject: it converts an analytic headache (multivaluedness) into clean geometry (a single-valued function on a nicer space). Every complete analytic function lives single-valuedly on its Riemann surface, continuation along a path becomes an ordinary walk on the surface, and deep questions about the function turn into topological questions about the surface — how many holes it has (its genus), how its sheets connect, whether it is a sphere or a torus or something with more handles. This is the bridge from analysis to topology and to algebraic geometry, since the Riemann surface of an algebraic function is precisely an algebraic curve.
Build the Riemann surface of sqrt(z) from two copies of the plane, each slit along the negative real axis. Glue the top edge of sheet 1's slit to the bottom edge of sheet 2's slit, and vice versa, with both joining at the branch point z = 0. Now sqrt(z) is single-valued: crossing the old cut moves you from sheet 1 (where sqrt = +...) onto sheet 2 (where sqrt = -...), exactly matching the sign flip you got from looping around 0.
Two slit planes glued crosswise along the cut, joined at z = 0: on this surface the square root is single-valued.
A Riemann surface is a two-real-dimensional surface, but a one-complex-dimensional manifold — locally it looks like a piece of the complex plane, not the real line. The branch cut is an artefact of how we drew the surface; on the surface itself there is no tear, the sheets join seamlessly.