Today I’m going to put some pieces of information on Riemann Mapping Theorem from across the web which I have been studying for a while. This post assumes a fundamental understanding of Conformal Mappings used in Complex Analysis. If not, this picture below should serve as an end-to-end introduction.

Another one:

Another one which makes it clearer:

I’m in no mood to cover Möbius Transformations so I’ll directly move to RMT.

So now let’s come back to RMT.

The Riemann mapping theorem states that if *U* is a non-empty simply connected (it just means that the domain *U* has no holes and is wholly connected within itself) open subset of the complex number plane **C** which is not all of **C**, then there exists a biholomorphic mapping *f* (i.e. a bijective holomorphic mapping whose inverse is also holomorphic) from *U* onto the open unit disk

The illustrations below further widened my outlook on RMT and its immense usefulness.

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I thereby conclude today’s post. I’ll come back with a new post sometime later in the week where I’ll cover further ramifications of a famous lemma which will kick start our discussion in a wholly new direction.

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