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3423/3424: Complex Analysis I/II
Lecturer: Dr. Dmitri Zaitsev
The course focuses around several main theorems, both proof (given below) and applications. There is on average four assignments between both semesters. The exam is worth a 100% of the course.
Outline of Riemann Mapping Theorem
Just learn the bold points for a minimal outline.
- Reduce to case is bounded, using fact is not all of
As , so on , so there is a square root , injective as . We can show there is such that but , let , then is bounded (by ).
- Create a family of injective maps taking to 0
Consider bounded, let injective . Let , there exists a sequence such that , obviously so uniformly bounded, so...
- Apply Montel's theorem to find a compactly convergent subsequence in converging to with the maximum possible
... Montel's theorem implies there is a compactly convergent subsequence tending to such that .
- Show is injective (using Rouche's theorem)
Assume not injective, let but , let , . Look at a disc not containing , suppose is non-zero on boundary of disc and note can choose so that (by uniform convergence of to ), Rouche then implies in disc, injectivity of then says in disc, contradiction.
- Show is surjective, by supposing not and then constructing a function satisfying
If no surjective, can find not in image of . Consider , non-zero on so has a square root , let , consider , sends to zero and is injective, an explicit calculation shows , a contradiction.