Riemann Mapping Theorem – The Crux of Complex Analysis


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

D = \{z\in \mathbf{C} : |z| < 1\}.

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

This slideshow requires JavaScript.

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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