# Questions tagged [riemann-sphere]

For questions about the Riemann sphere, a model of the extended complex plane.

44 questions
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### Lipschitz Mapping of $\mathbb{R}_+$ to $[0,1)$

During my research, I encountered a problem where I have some points in $\mathbb{R}^2_+$ (actually, there are in $\mathbb{N}^2$), and I would like to map them into the two-dimensional square $[0,1)^2$ ...
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### Is infinity the reciprocal of zero/is zero the reciprocal of infinity?

Is infinity the reciprocal of zero? Is zero the reciprocal of infinity? It would make sense that they would be--they behave in a similar way (anything multiplied by zero or infinity results in zero or ...
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### Covering space: preimage

Let $A = \{(x,y) \in \Bbb C^2: y^2 = f(x)\}$ where $f$ is a polynomial of degree $d$ without repeated roots. Let $f: A \to \Bbb C$ be defined by $f(x, y) = x$. For large $R$, what is the preimage ...
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### What are the mathematical properties of ⊥ in wheel theory?

I'm talking about the value of 0/0 in wheel theory, often denoted as "⊥." What are the behaviours of operations like addition, multiplication, exponentiation, trigonometric functions, their inverses, ...
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### How do you determine the distance from a projected point and its point of projection on a sphere having a unit radius

Let a sphere with a unit radius lie on the x-y plane, centered (touching) the point zero. Let a single point be somewhere on the surface of the sphere, but we can only see its projected point on the ...
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### What is the derivative of a polynomial at $\infty$?

Let $f$ be a polynomial defined on the Riemann sphere. I'm struggling to understand in what sense such a map can be said to be "holomorphic" at $\infty$. What is the derivative of $f$ at $\infty$? I ...
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### Conformal equivalence of finitely connected regions

Let us denote the open unit disk by $\Delta$ and write $\cong$ for conformal equivalence. Then the problem follows: Suppose that $G\subset\hat{\mathbb C}=\mathbb C\cup\{\infty\}$ is a region such ...
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### Meromorphic functions being continuous functions in the one point compactification of $C$

This is an exercise in Conway that I am stuck at. In fact I am surprised at this result. I cannot find a way to deal with the infinite points of $f$. Could anyone help me how to solve this exercise?
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### Roadmap to understand the link between spherical harmonics and Riemann sphere?

My ultimate goal is to see how the point of infinity and an arbitrary transform in Riemann sphere can lead to what consequences in dynamical systems, and it seems that harmonic analysis plays a ...
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### Is there any difference between a Riemann sphere and a normal stereographic projection of a complex plane?

As how I know, the Riemann sphere is a stereographic projection of the complex plane. Assuming that anyone who is familiar with projective geometry can have an intuition to do the same with complex ...
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### Has Fáry's theorem been proved/documented/given an exposition for embeddings into the 2-sphere?

This question is partly to breath new life into this question, which is essentially equivalent to the present one, partly to find a quotable reference quickly, since I am working on an exposition ...
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### Cross ratio on sphere

Project from the north pole to have an identification of the sphere in real space $\mathbb{R}^3$ and the complex projective line. Given $4$ (say, different) points on the sphere, I can project and ...
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### Rotation of the Riemann sphere [closed]

Show that the mapping $w = \frac{1 + z}{1 - z}$ corresponds to a $90$ degree counter-clockwise rotation of the Riemann sphere about the $y$-axis, $z$ is a complex number.
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### Uniformization of metrics vs. uniformization of Riemann surfaces

The uniformization theorem in complex analysis says that T1. Any Riemann surface of genus $0$ is conformally equivalent to the unit sphere. The uniformization theorem in differential geometry says ...
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### Non constant meromorphic function of Riemann Sphere

Let $X$ be a compact Riemann surface and $f$ a nonconstant meromorphic function on $X$. Show that $f$ must have a zero on $X$, and must have a pole on $X$. Suppose that $f$ is a nonconstant ...
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### Branch cut $\frac{\ln(z+i)}{1 +z^2}$

For each of the following functions $$\ln\frac{z−1}{ z+1},$$ $$\frac{\ln(z+i)}{1 +z^2},$$ $$\ln(z^2−1),$$ find location and order of the branch points, and give a valid branch cut. Can you help me ...
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### When mapping open disks D1 to D2 in the Riemann sphere would the boundaries (circles S1 and S2) be mapped to each other?

I've been struggling with a question for a few days now and I was hoping for some help, the question is as follows: f a homeomorphism of the Riemann sphere and D1 and D2 generalised disks with ...
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### Why is the complex domain of cosine naturally a sphere?

Near the end of this MAA piece about elliptic curves, the author explains why the complex domain of the cosine function is a sphere: since it's periodic, its domain can be taken as a cylinder, ...