كرة ريمان

The Riemann sphere can be visualized as the complex number plane wrapped around a sphere (by some form of stereographic projection – details are given below).

في الرياضيات ، كرة ريمان Riemann sphere ، على اسم الرياضي الشهير برنارد ريمان ، هي الطريقة الفريدة لإظهار السطح العقدي الممدد extended complex plane (السطح العقدي إضافة لنقطة في اللانهاية ) بحيث انه سيبدو من نقطة اللانهاية ممائلا لشكله عند أي عدد عقدي، بالذات بالنسبة للاستمرارية و الاشتقاقية. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value ∞ for infinity. With the Riemann model, the point "∞" is near to very large numbers, just as the point "0" is near to very small numbers.

The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as ${\displaystyle {\tfrac {1}{0}}=\infty }$ well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere.

In geometry, the Riemann sphere is the prototypical example of a Riemann surface, and is one of the simplest complex manifolds. In projective geometry, the sphere can be thought of as the complex projective line P1(C), the projective space of all complex lines in C2. As with any compact Riemann surface, the sphere may also be viewed as a projective algebraic curve, making it a fundamental example in algebraic geometry. It also finds utility in other disciplines that depend on analysis and geometry, such as the Bloch sphere of quantum mechanics and in other branches of physics.

The extended complex plane is also called closed complex plane.

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Extended complex numbers

The extended complex numbers consist of the complex numbers C together with ∞. The set of extended complex numbers may be written as C ∪ {∞}, and is often denoted by adding some decoration to the letter C, such as

${\displaystyle {\hat {\mathbb {C} }},\quad {\overline {\mathbb {C} }},\quad {\text{or}}\quad \mathbb {C} _{\infty }.}$

Geometrically, the set of extended complex numbers is referred to as the Riemann sphere (or extended complex plane).

Arithmetic operations

Addition of complex numbers may be extended by defining, for z ∈ C,

${\displaystyle z+\infty =\infty }$

for any complex number z, and multiplication may be defined by

${\displaystyle z\times \infty =\infty }$

for all nonzero complex numbers z, with ∞ × ∞ = ∞. Note that ∞ – ∞ and 0 × ∞ are left undefined. Unlike the complex numbers, the extended complex numbers do not form a field, since ∞ does not have a multiplicative inverse. Nonetheless, it is customary to define division on C ∪ {∞} by

${\displaystyle {\frac {z}{0}}=\infty \quad {\text{and}}\quad {\frac {z}{\infty }}=0}$

for all nonzero complex numbers z, with /0 = ∞ and 0/ = 0. The quotients 0/0 and / are left undefined.

ككرة

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 x2 + y2 + z2 = 1 in the three-dimensional real space R3. To this end, consider the stereographic projection 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 + iy. In Cartesian coordinates (x, y, z) and spherical coordinates (θ, φ) on the sphere (with θ the zenith and φ the azimuth), the projection is

${\displaystyle \zeta ={\frac {x+iy}{1-z}}=\cot \left({\frac {1}{2}}\theta \right)\;e^{i\phi }.}$

Similarly, stereographic projection from (0, 0, −1) onto the plane z = 0, identified with another copy of the complex plane by ξ = xiy, is written

${\displaystyle \xi ={\frac {x-iy}{1+z}}=\tan \left({\frac {1}{2}}\theta \right)\;e^{-i\phi }.}$

Automorphisms

A Möbius transformation acting on the sphere, and on the plane by stereographic projection
مقال رئيسي: Möbius transformation

The study of any mathematical object is aided by an understanding of its group of automorphisms, meaning the maps from the object to itself that preserve the essential structure of the object. In the case of the Riemann sphere, an automorphism is an invertible biholomorphic map from the Riemann sphere to itself. It turns out that the only such maps are the Möbius transformations. These are functions of the form

${\displaystyle f(\zeta )={\frac {a\zeta +b}{c\zeta +d}},}$

where a, b, c, and d are complex numbers such that adbc ≠ 0. Examples of Möbius transformations include dilations, rotations, translations, and complex inversion. In fact, any Möbius transformation can be written as a composition of these.

The Möbius transformations are homographies on the complex projective line. In projective coordinates, the transformation f can be written

${\displaystyle [\zeta ,\ 1]{\begin{pmatrix}a&c\\b&d\end{pmatrix}}\ =\ [a\zeta +b,\ c\zeta +d]\ =\ \left[{\tfrac {a\zeta +b}{c\zeta +d}},\ 1\right]\ =\ [f(\zeta ),\ 1].}$

Applications

In complex analysis, a meromorphic function on the complex plane (or on any Riemann surface, for that matter) is a ratio f/g of two holomorphic functions f and g. As a map to the complex numbers, it is undefined wherever g is zero. However, it induces a holomorphic map (f, g) to the complex projective line that is well-defined even where g = 0. This construction is helpful in the study of holomorphic and meromorphic functions. For example, on a compact Riemann surface there are no non-constant holomorphic maps to the complex numbers, but holomorphic maps to the complex projective line are abundant.

The Riemann sphere has many uses in physics. In quantum mechanics, points on the complex projective line are natural values for photon polarization states, spin states of massive particles of spin 1/2, and 2-state particles in general (see also Quantum bit and Bloch sphere). The Riemann sphere has been suggested as a relativistic model for the celestial sphere.[1] In string theory, the worldsheets of strings are Riemann surfaces, and the Riemann sphere, being the simplest Riemann surface, plays a significant role. It is also important in twistor theory.

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References

1. ^ R. Penrose (2007). The Road to Reality. Vintage books. pp. 428–430 (§18.5). ISBN 0-679-77631-1.
• Brown, James & Churchill, Ruel (1989). Complex Variables and Applications. New York: McGraw-Hill. ISBN 0-07-010905-2. Cite uses deprecated parameter |lastauthoramp= (help)
• Griffiths, Phillip & Harris, Joseph (1978). Principles of Algebraic Geometry. John Wiley & Sons. ISBN 0-471-32792-1. Cite uses deprecated parameter |lastauthoramp= (help)
• Penrose, Roger (2005). The Road to Reality. New York: Knopf. ISBN 0-679-45443-8.
• Rudin, Walter (1987). Real and Complex Analysis. New York: McGraw–Hill. ISBN 0-07-100276-6.

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