One of the most interesting ideas in complex analysis is called "analytic continuation. Suppose we start with a power series of form f(z) = ∑ a_n (z-c)^n where n = 0, 1, 2, ... and each a_n ∊ C and also c ∊ C. If this power series has a positive radius of convergence R, 0 < R ≤ oo, it defines the analytic function f(z) inside an open (circular) disk D_0 of radius R, centered at the complex number c. This can be viewed as shining a flashlight on the function to see how it's defined inside a certain disk. If R = oo, the function is defined for all z ∊ C and such an f(z) is called an "entire function". If R < oo, the power series won't converge for any z with |z - c| > R. (Its convergence for z exactly on the circle |z - c| = R can be quite complicated.) But if we try to use a power series with a different center, say c_1 ∊ D_0, we'll often find that the new disk of convergence, say D_1, extends beyond the boundary of D_0, thereby defining f(z) for some new values of z. Repeating this process, we can get a sequence of disks D_0, D_1, ..., D_k each overlapping the one before but extending beyond its boundary, defining f(z) on an expanding set. The sequence can be described just by its sequence of centers c = c_0, c_1, ..., c_k. It turns out that sometimes a new disk can overlap an old disk, however with values of f(z) that disagree for the same z. This happens notably for the function √(z-1) for a sequence of centers that goes around the point z = 1. Going once around, say picking centers lying on the circle C(t) = 1 + exp(it), 0 ≤ t ≤ 2π we find that the new value is the negative of the old value after traversing this circle once around. Indeed, going around a second time returns f(z) to its original values. We could take this kind of phenomenon to mean that f(z) = √(z-1) is a multi-valued function. Instead, Riemann's idea was to use this process to build a surface on which f(z) ends up being single-valued. The maximal surface on which this is possible turns out to be uniquely defined, and came to be called the "Riemann surface of f". In this overly simplified description, we've overlooked how some "defects" in the function and its Riemann surface can be "fixed". One is that, initially, a point z_1 where |f(z)| —> oo as z —> z_1 is thought of as a singularity of f, known as a "pole". The function f(z) = 1/z^3 for instance has a pole at z = 0. But we can add the point named "oo" to the complex plane, obtaining its one-point compactification C u {oo} sometimes called the "Riemann sphere" because, topologically, it is indeed homeomorphic to the sphere S^2 and we may think of it as just being S^2. (A convenient way to go back and forth between S^2 and C u {oo} is stereographic projection.) A second kind of "defect" is the branch point, such as f(z) = √(z-1) has at the point z = 1. This is any point for which going around it along an arbitrarily small circle and coming back to where you began results in a new value of the function. There are finite branch points like those of √(z-1), where enough circumnavigation will return the function to its original values. But there also exist branch points of infinite order, like that of f(z) = log(z), where no amount of going around the branch point gets you back to the function's original values. The latter kind can't be "fixed". But a finite branch point can be patched up so that the function's domain extends over the patch in an analytic way. Thus, every power series initially defined only inside its radius of convergence ends up defining a unique maximal domain of definition of the function, its "Riemann surface". A Riemann surface can be any smooth surface (that's Hausdorff and has a countable base to its topology). Every such has a well-defined universal covering space that's simply connected (i.e., every closed curve can be shrunk to a point). One of the greatest theorems in complex analysis is the Uniformization Theorem of Poincaré and Koebe, which proves that a simply connected Riemann surface can be just one of three possibilities, up to conformal equivalence: * the sphere S^2, * the complex plane C, or * the (open) unit disk D. Where "conformal equivalence between two Riemann surfaces" means there is a smooth, angle-preserving homeomorphism between them. This implies that *every* Riemann surface is the result of taking the quotient of one of these three simply connected surfaces by a discrete group acting on it, without fixed points and what's called "properly discontinuously". (Explained here if you're curious: <https://www.math.ucdavis.edu/~kapovich/EPR/prop-disc.pdf <https://www.math.ucdavis.edu/~kapovich/EPR/prop-disc.pdf>>.) —Dan