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? 
 A: 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.
A: 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.
A: 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.
A: 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.
A: The uniformization/Riemann mapping theorem should give you lots to chew on. 
A: 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. 
A: You may find some inspiration in Needham's Visual Complex Analysis.
A: 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. 
A: I feel that complex function and color domain are interesting topics. I was thinking about its application in certain software. For example Blender is one of them that uses the model RGB for designing complex diagrams…
