Multiple-valued complex functions {Riemann surface} can be complex-plane spirals {winding space, spiral}.
point
Riemann surfaces have a central point {branch point} about which to turn.
infinity
Surfaces can rejoin after a finite number of turns {finite order, surface} or can be infinite.
logarithm function
Logarithm-function Riemann surface is not compact but can compact to Riemann sphere.
Riemann sphere
The simplest compact/closed Riemann surface {Riemann sphere} has complex plane goes through equator. Complex plane stereographically projects onto one hemisphere, and its reciprocal projects onto other hemisphere. Circles or straight lines on complex planes are circles on spheres.
genus
Sphere has genus 0, because it has no complex moduli and has three holomorphic self-transformation parameters, for bilinear transformations. Torus has genus 1, because it has one complex modulus and one holomorphic self-transformation parameter, for translation. Genus 2 has three complex moduli and no holomorphic self-transformation parameters. Genus n has 3*n - 3 complex moduli and no holomorphic self-transformation parameters.
Mathematical Sciences>Algebra>Function>Kinds>Complex
3-Algebra-Function-Kinds-Complex
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Date Modified: 2022.0224