Riemann Sphere Complex Analysis

The Riemann sphere can be visualized as the complex number plane wrapped around a sphere (by some form of – details are given below).In, the Riemann sphere (or extended complex plane), named after the 19th century mathematician, is the obtained from the by adding a. The sphere is the geometric representation of the extended complex numbers, which consist of the together with a symbol to represent.The extended complex numbers are useful in because they allow for in some circumstances, in a way that makes expressions such as. For example, any on the complex plane can be extended to a on the Riemann sphere, with the of the rational function mapping to infinity. More generally, any can be thought of as a continuous function whose is the Riemann sphere.In, the Riemann sphere is the prototypical example of a, and is one of the simplest. In, the sphere can be thought of as the complex, the of all in. As with any Riemann surface, the sphere may also be viewed as a projective, making it a fundamental example in. It also finds utility in other disciplines that depend on analysis and geometry, such as and other branches of.Extended complex numbersThe extended complex numbers consist of the complex numbers together with.

Leads To: MA475 Riemann Surfaces. Content: The course focuses on the properties of differentiable functions on the complex plane. Unlike real analysis, complex differentiable functions have a large number of amazing properties, and are very ``rigid' objects. Some of these properties have been explored already in Vector Analysis.

The extended complex numbers may be written as, and are often denoted by adding some decoration to the letter, such asGeometrically, the set of extended complex numbers is referred to as the Riemann sphere (or extended complex plane). Arithmetic operationsof complex numbers may be extended by definingfor any complex number, and may be defined byfor all nonzero complex numbers, with.

Note that and are left undefined. Unlike the complex numbers, the extended complex numbers do not form a, since does not have a. Nonetheless, it is customary to define on byfor all nonzero complex numbers, with. Rational functionsAny f( z) = g( z) / h( z) can be extended to a on the Riemann sphere.

Specifically, if z 0 is a complex number such that the denominator h( z 0) is zero but the numerator g( z 0) is nonzero, then f( z 0) can be defined as. (If both the numerator and denominator are zero, then they share a common factor, and the fraction should first be reduced to lowest terms.) Moreover, can be defined as the of f( z) as, which may be finite or infinite.For example, given the functionwe may define since the denominator is zero at, and since as. Using these definitions, becomes a continuous function from the Riemann sphere to itself.When viewed as a complex manifold, these rational functions are in fact from the Riemann sphere to itself. As a complex manifoldAs a one-dimensional complex manifold, the Riemann sphere can be described by two charts, both with domain equal to the complex number plane. Let ζ and ξ be complex coordinates on. Identify the nonzero complex numbers ζ with the nonzero complex numbers ξ using the transition mapsSince the transition maps are, they define a complex manifold, called the Riemann sphere.Intuitively, the transition maps indicate how to glue two planes together to form the Riemann sphere. The planes are glued in an 'inside-out' manner, so that they overlap almost everywhere, with each plane contributing just one point (its origin) missing from the other plane.

In other words, (almost) every point in the Riemann sphere has both a ζ value and a ξ value, and the two values are related by ζ = 1 / ξ. The point where ξ = 0 should then have ζ-value ' 1 / 0'; in this sense, the origin of the ξ-chart plays the role of ' ' in the ζ-chart. Symmetrically, the origin of the ζ-chart plays the role of in the ξ-chart., the resulting space is the of a plane into the sphere. However, the Riemann sphere is not merely a topological sphere. It is a sphere with a well-defined, so that around every point on the sphere there is a neighborhood that can be identified with.On the other hand, the, a central result in the classification of Riemann surfaces, states that the only simply-connected one-dimensional complex manifolds are the complex plane, the, and the Riemann sphere.

The best place to get cheats, codes, cheat codes, walkthrough, guide, FAQ, unlockables, trophies, and secrets for Warriors Orochi 3 Ultimate for PlayStation 4 (PS4). Warriors orochi 3 ultimate ps vita cheats. This page contains a list of cheats, codes, Easter eggs, tips, and other secrets for Warriors Orochi 3 Ultimate for Xbox One.If you've discovered a cheat you'd like to add to the page, or have a. For Warriors Orochi 3 Ultimate on the PlayStation 4, GameFAQs has 475 cheat codes and secrets. The best place to get cheats, codes, cheat codes, walkthrough, guide, FAQ, unlockables, achievements, and secrets for Warriors Orochi 3 Ultimate for Xbox One.

Of these, the Riemann sphere is the only one that is a (a surface without boundary). Hence the two-dimensional sphere admits a unique complex structure turning it into a one-dimensional complex manifold.

As the complex projective lineThe Riemann sphere can also be defined as the complex projective line. This is the subset of consisting of all pairs (α,β) of complex numbers, not both zero, modulo the (α,β) = (λα,λβ)for all nonzero complex numbers λ. The complex plane, with coordinate ζ, can be mapped into the complex projective line by (α,β) = (ζ,1).Another copy of with coordinate ξ can be mapped in by (α,β) = (1,ξ).These two complex charts cover the projective line. For nonzero ξ the identifications (1,ξ) = (1 / ξ,1) = (ζ,1)demonstrate that the transition maps are ζ = 1 / ξ and ξ = 1 / ζ, as above.This treatment of the Riemann sphere connects most readily to projective geometry.

For example, any line (or smooth conic) in the is biholomorphic to the complex projective line. It is also convenient for studying the sphere's, later in this article. Stereographic projection of a complex number A onto a point α of the Riemann sphere.The Riemann sphere can be visualized as the unit sphere x 2 + y 2 + z 2 = 1 in the three-dimensional real space. To this end, consider the from the unit sphere minus the point (0,0,1) onto the plane z = 0, which we identify with the complex plane by ζ = x + i y. In Cartesian coordinates ( x, y, z) and spherical coordinates (ϕ,θ) on the sphere (with ϕ the zenith and θ the azimuth), the projection isSimilarly, stereographic projection from (0,0, − 1) onto the z = 0 plane, identified with another copy of the complex plane by ξ = x − i y, is writtenIn order to cover the unit sphere, one needs the two stereographic projections: the first will cover the whole sphere except the point (0,0,1) and the second except the point (0,0,-1).

Hence, one needs two complex planes, one for each projection, which can be intuitively seen as glued back-to-back at z=0. Note that the two complex planes are identified differently with the plane z = 0.

An -reversal is necessary to maintain consistent orientation on the sphere, and in particular complex conjugation causes the transition maps to be holomorphic.The transition maps between ζ-coordinates and ξ-coordinates are obtained by composing one projection with the inverse of the other. They turn out to be ζ = 1 / ξ and ξ = 1 / ζ, as described above. Thus the unit sphere is to the Riemann sphere.Under this diffeomorphism, the unit circle in the ζ-chart, the unit circle in the ξ-chart, and the equator of the unit sphere are all identified. The unit disk ζ 0.

MetricA Riemann surface does not come equipped with any particular Riemannian metric. However, the complex structure of the Riemann surface does uniquely determine a metric up to. (Two metrics are said to be conformally equivalent if they differ by multiplication by a positive.) Conversely, any metric on an oriented surface uniquely determines a complex structure, which depends on the metric only up to conformal equivalence.

Complex structures on an oriented surface are therefore in one-to-one correspondence with conformal classes of metrics on that surface.Within a given conformal class, one can use conformal symmetry to find a representative metric with convenient properties. In particular, there is always a complete metric with in any given conformal class.In the case of the Riemann sphere, the Gauss-Bonnet theorem implies that a constant-curvature metric must have positive K. Games to checkout on steam. It follows that the metric must be to the sphere of radius in via stereographic projection. In the ζ-chart on the Riemann sphere, the metric with K = 1 is given byIn real coordinates ζ = u + i v, the formula isUp to a constant factor, this metric agrees with the standard Fubini–Study metric on complex projective space (of which the Riemann sphere is an example).Conversely, let S denote the sphere (as an abstract or ).

By the uniformization theorem there exists a unique complex structure on S. It follows that any metric on S is conformally equivalent to the round metric. All such metrics determine the same conformal geometry. The round metric is therefore not intrinsic to the Riemann sphere, since 'roundness' is not an invariant of conformal geometry. The Riemann sphere is only a conformal manifold, not a. However, if one needs to do Riemannian geometry on the Riemann sphere, the round metric is a natural choice.