Final year project ideas - complex analysis

For my final year, I have to do a project for a module. I want to investigate something in the complex analysis area. I've only covered the basics of analysis, like Cauchy's IT/IF, residue theorem etc. The only thing that's been suggested so far is the mathematics of Aerofoils. Just wondering if anyone has ideas of areas I could look at?

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Converted to a Wiki since it is asking for a list of ideas. – Willie Wong Mar 19 '13 at 16:14

I suggest you to look at some special functions, i.e., gamma function, zeta function and theta function. And the connection between cplx analysis and number theory. It's really interesting that the property of these functions will lead to some property of prime numbers. You can especially focus on the fundamental theorem of prime numbers and the four square theorem.

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I've not studied those before, I'll take a look. Thanks! – Mike Miller Mar 19 '13 at 13:49
I suggest you to take a look at the book of E.M.Stein. – lee Mar 19 '13 at 14:20
... and Rami Shakarchi; there are two authors to that book. – Willie Wong Mar 19 '13 at 16:16
@BritMiller: Dear Brit, The book Riemann's zeta function, by Edwards, gives a very nice exposition of the zeta-function, and related topics such as the Gamma function. It also discusses a lot of interesting topics in analysis that you don't often see in courses, such as Mellin transforms, Euler--Maclaurin summation, and asymptotic expansions. Regards, – Matt E Mar 19 '13 at 16:43
Thank you all, I'll take a look. – Mike Miller Mar 19 '13 at 17:38

The uniformization/Riemann mapping theorem should give you lots to chew on.

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If you can read French, don't miss the book Uniformisation des surfaces de Riemann - Retour sur un théorème centenaire. The PDF is freely available. – lhf Mar 19 '13 at 16:34
See also mathoverflow.net/questions/10516/…. – lhf Mar 19 '13 at 16:36
These are exactly the sort of things I was looking for; thank you! – Mike Miller Mar 19 '13 at 17:31
@B11: if you have another question, please use the Ask Question link near the top right of the page. – robjohn Mar 20 '13 at 3:28

I would suggest studying Fourier and Laplace transforms. Learning them well will suit you in the future, regardless of which field you end up in.

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There is a tutor who runs a project on both of these, but she said they weren't particularly challenging and generally for people aiming for a 2:1; she suggested I try something more challenging. – Mike Miller Mar 19 '13 at 13:28

The theory of minimal surfaces is kind of interesting, there are a lot of nice pictures out there of the strange shapes you can make. Learning about the Poisson kernel and seeing how it works is interesting too.

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Thank you, I will take a look. – Mike Miller Mar 19 '13 at 13:33

I always found the Schwarz-Christoffel map to be particularly fascinating and always an area of active interest. The applications are boundless, but really they are used for solving flow-type problems around awful, polygonal boundaries.

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Excellent, that looks very interesting. – Mike Miller Mar 19 '13 at 13:48

A slightly less well-known example is the actual proof of the Runge phenomenon. It is one of my favourite applications of the residue theorem. You can find a sketch of the proof in this document with links to other resources.

In the same vein, Lax and Zalcman's book contains a lot of very interesting examples that one usually do not encounter during a first course in complex analysis.

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Thank you, I've not seen the proof, I'll read through it. – Mike Miller Mar 19 '13 at 17:37

You may find some inspiration in Needham's Visual Complex Analysis.

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Library has it in stock, I'll take a look, thanks! – Mike Miller Mar 20 '13 at 11:51

Whatever you choose, make sure it really interests you. You will spend a lot of time on the subject, if you hate it after a week or so, you are in trouble... Pick something reasonably wide, so you have space to move around.

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Yeah, I was thinking of picking something I could carry on at MSc level (if I manage to get funding!), so going to spend the next couple of months researching different areas. – Mike Miller Mar 19 '13 at 19:29