What is a Riemann sphere used for?

What is a Riemann sphere used for?

noun Mathematics. a sphere used for a stereographic projection.

Is the Riemann sphere a field?

I think its most natural algebraic interpretation the Riemann sphere has is its identification with the complex projective line, so that it is a set upon which the Möbius group acts upon. So, not a field, ring or algebra, but the underlying space of a group action.

What is stereographic projection in complex analysis?

In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. It is conformal, meaning that it preserves angles at which curves meet.

What is the extended complex plane?

The extended complex plane is the name gives to the complex plane with a point at infinity attached: , where denotes complex infinity. It is also called the Riemann sphere and is various denoted or .

What is complex infinity?

Complex infinity is an infinite number in the complex plane whose complex argument is unknown or undefined. Complex infinity may be returned by the Wolfram Language, where it is represented symbolically by ComplexInfinity. The Wolfram Functions Site uses the notation. to represent complex infinity.

Why are Riemann surfaces important?

The main interest in Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm.

What is the purpose of stereographic projection?

Stereographic projection is a technique for displaying the angular properties of a plane faced object on a single drawing or diagram. Directions as well as planes may be shown and any desired angle can be measured directly from the projection using a graphical technique.

Why do we use stereographic projection?

Stereographic projection is a method used in crystallography and structural geology to depict the angular relationships between crystal faces and geologic structures, respectively. Here we discuss the method used in crystallography, but it is similar to the method used in structural geology.

When Infinity is adjoined to the complex plane the plane is termed?

We perfect the one-to-one correspondence by adding one more point to the complex plane – the so-called point at infinity – and identifying it with the north pole on the sphere. This topological space, the complex plane plus the point at infinity, is known as the extended complex plane.

Why do we call ourselves the Riemann sphere?

When we saw the real and complex numbers didn’t allow us to divide by zero without contradictions, we simply made our own structure that would allow us to do so. This is why we called ourselves the Riemann Sphere. Aside from the fact that it contains everything from zero to infinity inclusive, it represents the freedoms mathematics gives us.

Is the Riemann sphere a meromorphic function of the complex plane?

In this context, meromorphic functions to the complex plane are holomorphic functions to the Riemann sphere. Topologically, the Riemann sphere is indeed a sphere, so the term is justified. Indeed it is the one-point compactification of the complex plane.

What is Carl Riemann’s model of the complex plane?

Carl created a model of the complex plane where division by zero IS defined, and named it after the great father of complex analysis Bernard Riemann, of the famed Riemann Hypothesis. There’s a bunch of complex equations out there describing how to construct it, but let’s take a look at a more intuitive way of thinking about it.

What is a homeomorphism from the sphere S to the complex plane?

The line corresponding to 0 will intersect the sphere at the origin. Every other point on the plane will have a unique point on the sphere corresponding to it in this way. This is what we mean by a homeomorphism from the sphere S to the complex plane C . Think about that for a moment, though.