Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [riemann-sphere]

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

0
votes
0answers
10 views

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 $[0,1)^2$ ...
2
votes
1answer
80 views

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 ...
2
votes
0answers
32 views

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 ...
1
vote
1answer
55 views

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, ...
0
votes
1answer
35 views

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 ...
0
votes
0answers
50 views

Is there a “natural” entire series associated to the Riemann zeta function whose radius of convergence is $\frac{1}{\sqrt{5}}$?

As a follow-up to Is there a hidden connection between RH and the golden ratio?, let's consider the plane $ P $ whose intersection with the Riemann sphere is the circle I denoted by $ \Gamma_{\Delta}...
0
votes
0answers
34 views

Show that a polynom $P$ is continuable to a holomorphic function $\hat{P}:P^1\mathbb{C}\to P^1\mathbb{C}$.

The polynom is defined as $P(z)=a_n*z^n+...+a_0$ with degree n. $P^1\mathbb{C}$ ist the projective line. An analytic continuation is defined as: Suppose f is an analytic function defined on a ...
2
votes
2answers
233 views

What is the real and imaginary part of complex infinity?

I know that complex infinity is a pole on the Riemann-sphere, but what is actually the real and imaginary part of it? Is it $\pm\infty \pm \infty i$? And how is it different from simple non-complex ...
0
votes
1answer
42 views

How to describe the relative positions?

There is a problem in my complex variable textbook as follows: Discribe the relative positions of the images of $z$, $-z$ and $\bar z$ on the Riemann sphere. But I don't understand what does this ...
1
vote
0answers
206 views

Residue Theorem: Inside vs. Outside

My lecturer told me that when I use the residue theorem, it does not matter if I sum over the residues of the poles on the inside or the outside of a positively oriented simple closed curve $\gamma$. ...
2
votes
1answer
96 views

Riemann sphere and Fundamental theorem of algebra

Recently, I saw the following proof of the Fundamental theorem of algebra through the Gauss-Bonnet theorem. https://arxiv.org/pdf/1106.0924.pdf In here, $\mathbb{C} \cup \{\infty\}$ is identified to $\...
2
votes
1answer
45 views

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 ...
0
votes
1answer
91 views

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 ...
0
votes
1answer
107 views

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?
3
votes
0answers
251 views

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 ...
0
votes
0answers
252 views

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 ...
1
vote
0answers
23 views

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 ...
3
votes
1answer
100 views

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 ...
-1
votes
2answers
111 views

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.
4
votes
1answer
196 views

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 ...
1
vote
1answer
78 views

Metric on a Quotient of the Riemann Sphere (Revised) [duplicate]

Let $P$ denote the quotient space obtained by the action of $\mathbb{Z}\backslash2\mathbb{Z}$ by the map $z\mapsto\frac{1}{z}$ on the riemann sphere $\hat{\mathbb{C}}$ (identified here with $\mathbb{C}...
1
vote
4answers
231 views

Metric on a Quotient of the Riemann Sphere

Let $P$ denote the quotient space obtained by the action of $\mathbb{Z}\backslash2\mathbb{Z}$ by the antipodal map $z\mapsto\frac{1}{z}$ on the riemann sphere $\hat{\mathbb{C}}$ (identified here with $...
0
votes
0answers
233 views

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 ...
1
vote
0answers
107 views

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 ...
2
votes
0answers
68 views

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 ...
6
votes
2answers
274 views

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, ...
1
vote
1answer
90 views

Approximating volume of a sphere using spherical cones in the limit

[Disclaimer: I am not a mathematician, please bear my naive wordings] I am looking for a reference that can give me a proof to my simple question: Can we approximate the volume of a sphere using ...
1
vote
0answers
230 views

Cohomology of the Riemann sphere

Let us note $\overline{\mathbb{C}}$ the Alexandroff compactification of $\mathbb{C}$ (i.e. the Riemann Sphere). I can prove that $$H^1(\overline{\mathbb{C}},\Omega_{\overline{\mathbb{C}}}) \simeq \...
1
vote
0answers
85 views

Exercise of Rick Miranda is wrong? Actions over Riemann sphere

I'm studying the book Rick Miranda, Algebraic Curves and Riemann Surfaces and I have a question about the exercise H of page 84. The book says that $z \mapsto exp(2\pi i /r)z$ is an automorphism of ...
0
votes
0answers
47 views

What's the official name for a space that warps like this?

For years, games have had this idea of a space that warps. Attempting to move out of the playing field on the left side would send you to the right side, like in Pac-Man. But is there an official ...
6
votes
1answer
152 views

Polyhedral symmetry in the Riemann sphere

I found an interesting set of constants while exploring the properties of nonconstant functions that are invariant under the symmetry groups of regular polyhedra in the Riemann sphere, endowed with ...
10
votes
4answers
1k views

The Riemann Sphere Interpretation

Is the Riemann sphere anything more than a simple visual tool to help students understand the complex planes, or the behavior of complex valued functions at infinity, limit points etc? Or is there a ...
0
votes
1answer
313 views

Branched coverings of the Riemann sphere

Can someone give me an example of a non-trivial branched covering of the Riemann sphere? Is there some way to enumerate all such coverings? Is there any easy answer to the same questions about the ...
2
votes
0answers
59 views

Open sets in $\mathbb C$ and open sets in $\hat{\mathbb C}$

I usually have a lot of trouble with complex variable when it comes to the geometric representation of $\hat{\mathbb C}$ and what happens in there. I have the next exercise and from quick look at it I ...
0
votes
1answer
301 views

Why is the Z-transform of $e^{at}$, t = kT, different from Laplace transform of $e^{at}$

The Laplace transform of $e^{at}$ takes a well known form of $\frac{1}{s-a}$ The Z transform of $e^{at} = e^{akT} $ T is the sampling period takes the form of $\frac{z}{z-e^{aT}}$ Does anyone know ...
2
votes
1answer
473 views

Intersection of a Plane with the Riemann Sphere

While reading Fundamentals of Complex Analysis by Saff and Snider, I came across an example (see page 47, edition 3) where it is shown that "all lines and circles in the $z$-plane correspond under ...
2
votes
1answer
58 views

Is it possible to build a metal wire sphere whose shadow projects the 2d conformal maps of the Riemann Sphere on a flat wall?

Is it possible to build a metal wire hollow sphere whose shadow from a nearby point light source projects the 2d conformal maps of the Riemann Sphere on a flat wall? I believe that in theory it ...
1
vote
1answer
95 views

What is this metric called?

Ahlfors -complex analysis p.20 Consider a stereographic projection between the 2-sphere and $\overline{\mathbb{C}}$ (i.e. one-point compactification of $\mathbb{C}$) Let $z,w$ be complex numbers. ...
1
vote
0answers
72 views

What “topological setting” would make complex analysis fluent?

In measure theory, the order topology on $\overline{\mathbb{R}}$ (extended real) and $[0,\infty]$ provides rich foundation to analyze measurable functions and abstract integral. Just like this, i ...
1
vote
2answers
331 views

Adding infinity to the upper half plane

I have a weak physicist background in complex analysis and topology. I've been looking at things defined on the upper half complex plane, and it is not clear to me if there are subtleties in "going to ...
4
votes
0answers
738 views

A domain on a sphere is simply connected if and only if its complement is connected

I think the statement that a domain (open connected set) in a sphere is simply connected if and only if its complement is connected is a standard result. But how can one prove it? Is it possible to ...
0
votes
1answer
80 views

Integrate a function over the union sphere (Stokes)?!

I have problems solving this integral since 2 days -.- $\int_{S^2} f \cdot n \; dS \; \; \;$ with $f(x,y,z):=(y^3, z^3, x^3)^T$, $\; \; S^2=\{(x,y,z) \in \mathbb{R^3} | x^2+y^2+z^2 = 1\}$ and $n$ ...
3
votes
1answer
436 views

When is $\infty$ a critical point of a rational function on the sphere?

In his "Iteration of Rational Functions" Beardon defines a critical point of a rational function (mapping the Riemann sphere to itself) as a point such that the function is not injective in any ...
3
votes
0answers
114 views

Characterizations of a linear fractional transformation

Consider the function $$ g(t) = \frac{1+it}{1-it} = \frac{1-t^2}{1+t^2} + i \frac{2t}{1+t^2}. $$ (The second equality holds except when $t=i$.) It seems to be widely known that this function is the ...