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2 votes
0 answers
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Convergence of finite-difference method for Cauchy-Riemann equations

Let $I\subseteq \mathbb{R}$ an open interval. Let $f:I\rightarrow \mathbb{C}$ real analytic. Suppose we want to numerically compute an analytic extension of $f$. We will assume the following: we are ...
Plemath's user avatar
  • 312
0 votes
0 answers
66 views

Uniformization and constructive analytic continuation of Taylor-Maclaurin series

Context. In their paper, "Uniformization and Constructive Analytic Continuation of Taylor Series", Costin and Dunne present a constructive method to greatly increase the accuracy of a ...
butsurigakusha's user avatar
3 votes
0 answers
114 views

Conformal welding and Jordan loop consequences?

In the similar context as Conformal welding of rectifiable curves In classical conformal welding theory, we start with a homeomorphism $h$ of the unit circle and try to find a Jordan domain $D$ ...
Thomas Kojar's user avatar
  • 5,474
4 votes
1 answer
451 views

Riemann mapping theorem with smooth boundary

This is closely related to the question here. The setup is that $U\subset\mathbb{C}$ is an open bounded simply connected domain with $C^\infty$ boundary. If $\phi:U\rightarrow\mathbb{D}$ is a ...
J_P's user avatar
  • 439
3 votes
0 answers
90 views

Boundary behavior of conformal map on domain satisfying an exterior sphere condition

I'm in the middle of a project concerning a Bernoulli-type free boundary problem in $\mathbb{R}^2$ and, as part of this project, I would like to understand the boundary behavior of conformal maps on ...
Gary Moon's user avatar
  • 683
11 votes
3 answers
748 views

Explicit triples of isomorphic Riemann surfaces

Inspired by a discussion with Neil Strickland I am very interested to hear of explicit examples (one per answer, please), as follows. A compact Riemann surface can be presented in many different ways....
2 votes
2 answers
2k views

What is a simplified intuitive explanation of conformal invariance? [closed]

Can the concept of conformal map and conformal Invariance be explained in very general terms, preferably in high school/undergrad-level Mathematics? Abstracting away from the applications in physics (...
Sohail Si's user avatar
  • 157
0 votes
1 answer
576 views

Existence of an inverse to the Schwarz-Christoffel mapping [closed]

As an elementary result in complex analysis, one can use the argument principle to show that the Schwarz-Christoffel transform is injective on the interior of the polygon to which it maps. Could this ...
Talmsmen's user avatar
  • 547
3 votes
1 answer
590 views

Reference on boundary behavior of conformal maps

I am looking for some results on the boundary behavior of conformal maps between simply connected domains. In particular I am interested in conformal maps between $\mathbb{C}-\Delta$, where $\Delta$ ...
Luis Giraldo Gonzalez's user avatar
15 votes
1 answer
2k views

How to interpret Gauss's late fragments on conformal mapping of the interior of an ellipse (to the unit disk) in modern mathematical terms?

My question refers to some not very well known (and unpublished) fragments of Gauss that treat the problem of finding a conformal mapping (angle-preserving mapping) in the complex plane from the ...
user2554's user avatar
  • 2,099
24 votes
3 answers
3k views

How to find a conformal map of the unit disk on a given simply-connected domain

By the classical Riemann Theorem, each bounded simply-connected domain in the complex plane is the image of the unit disk under a conformal transformation, which can be illustrated drawing images of ...
Taras Banakh's user avatar
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6 votes
1 answer
468 views

Factorization of conformal maps between annuli

Consider two doubly-connected open subsets $A$ and $A'$ of the Riemann sphere. We assume these two domains to be of same modulus (the moduli space being one real parameter), i.e. we assume that there ...
Stéphane Benoist's user avatar
4 votes
2 answers
2k views

Non-bijective conformal maps between annuli

I need to answer the following question, hopefully in the negative. Question: Does there exist a conformal map $f$ of degree $1$ from the annulus $\{1<|z|<R\}$ to the punctured disk $\{0<|...
Xin Nie's user avatar
  • 1,804
0 votes
1 answer
405 views

Riemann mapping

Let in the complex plane be a bounded Jordan region T (that is a bounded and simply connected set with the boundary a Jordan curve), containing the origin, with its Riemann mapping onto the open unit ...
George's user avatar
  • 71
8 votes
1 answer
2k views

Is there a manifold structure on a space of conformal maps?

I would be very grateful for any information or pointers for the following: 1) Fix an open subset $U$ of $\mathbb{CP}^1$. a) Does the set of all holomorphic maps from $U$ to $\mathbb{C}$ (with the ...
Thomas K's user avatar
12 votes
1 answer
5k views

Conformal maps of doubly connected regions to annuli.

In another question here on MO, Anweshi asks if any doubly connected region in the complex plane can be conformally mapped to some annulus. The answer to this is yes. But the fact is that two annuli ...
GMRA's user avatar
  • 2,050
8 votes
3 answers
2k views

Riemann mapping for doubly connected regions

Remove the closure of simply connected region from the interior of a simply connected region. Is it true that the resulting domain can be mapped conformally to some annulus?
Anweshi's user avatar
  • 7,442