C-element

simply connected Julia set as an image of unit circle under Riemann map

In mathematical complex analysis, Carathéodory's theorem, proved by Template:Harvtxt, states that if U is a simply connected open subset of the complex plane C, whose boundary is a Jordan curve Γ then the Riemann map

f: U → D

from U to the unit disk D extends continuously to the boundary, giving a homeomorphism

F : Γ → S¹

from Γ to the unit circle S¹.

Such a region is called a Jordan domain. Equivalently, this theorem states that for such sets U there is a homeomorphism

F : cl(U) → cl(D)

from the closure of U to the closed unit disk cl(D) whose restriction to the interior is a Riemann map, i.e. it is a bijective holomorphic conformal map.

Another standard formulation of Carathéodory's theorem states that for any pair of simply connected open sets U and V bounded by Jordan curves Γ₁ and Γ₂, a conformal map

f : U→ V

extends to a homeomorphism

F: Γ₁ → Γ₂.

This version can be derived from the one stated above by composing the inverse of one Riemann map with the other.

A more general version of the theorem is the following. Let

g : D$\to$ U

be the inverse of the Riemann map, where D ⊂ C is the unit disk, and U ⊂ C is a simply connected domain. Then g extends continuously to

G : cl(D) → cl(U)

if and only if the boundary of U is locally connected. This result was first stated and proved by Marie Torhorst in her 1918 thesis, under the supervision of Hans Hahn, using Carathéodory's theory of prime ends.

Context

Intuitively, Carathéodory's theorem says that compared to general simply connected open sets in the complex plane C, those bounded by Jordan curves are particularly well-behaved.

Carathéodory's theorem is a basic result in the study of boundary behavior of conformal maps, a classical part of complex analysis. In general it is very difficult to decide whether or not the Riemann map from an open set U to the unit disk D extends continuously to the boundary, and how and why it may fail to do so at certain points.

While having a Jordan curve boundary is sufficient for such an extension to exist, it is by no means necessary . For example, the map

f(z) = z²

from the upper half-plane H to the open set G that is the complement of the positive real axis is holomorphic and conformal, and it extends to a continuous map from the real line R to the positive real axis R⁺; however, the set G is not bounded by a Jordan curve.

References

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Further References

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
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