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I was interested in doing some research in complex analysis. I already have a basic understanding of the subject. i.e. I read Saff and Snider's book "Fundamentals of Complex Analysis". But now I would like to write a paper in my free time for fun. But I have no idea whatsoever what a beginner in complex analysis could do a paper on. So I was wondering if anyone had any suggestions as to a topic that I could research more in depth and write a paper on. Thanks :)

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@BillyKendrick Contact the mathematics department at your local college and ask if there is anyone there who would be able to supervise a project suitable for science fairs like the Intel Talent Search and the Westinghouse competition. It is fairly common for professors to take talented high schoolers under their wing and supervise such projects. Give your background (mainly the lack of a rigorous course in analysis), I doubt you could do research in complex analysis. However, if you simply wanted to learn about it, a professor could facilitate that too. –  Potato Jan 16 '13 at 2:52
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No one is saying that you can't do research in the field, but complex analysis is a vast subject and it's hard for someone with just an intro to do substantial research and produce results. Especially if the deadline is in 11 days - that's not at all realistic to expect to produce a research paper from scratch in 11 days, in any field. My advice is, if you want to do publishable research in complex analysis, then learn as much mathematics (complex and otherwise) as you can - you're a long way from the frontier of research. Consult a professor to see how exactly you should go about this. –  kigen Jan 16 '13 at 3:07
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@BillyKendrick Ten days is an extremely short amount of time, and almost certainly not enough time to do original mathematics research worthy of entrance in a science fair. I'm not trying to be mean, but it must be said. Regarding my comment about you being unlikely to do research: I mean this not as an insult to your abilities, but as a remark about mathematics research in general. Much is already known about complex analysis, and even graduate students who have completed Ph.D. qualifying courses in it would find it impossible to produce new work without significant further study. –  Potato Jan 16 '13 at 3:09
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@BillyKendrick, don't feel bad: you really can't do research, as understood in universities, in analysis taking into account what you said you've studied, and yes: perhaps it is a little too late for you to begin thinking of writing a paper that's supposed to have a level above high school stuff if you knew you had to present it by January 26-th... –  DonAntonio Jan 16 '13 at 3:10
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There seems to be a lot of misunderstanding on this thread. Let's clarify. Everyone: Billy is a high school student who has self-studied complex analysis. He wants to write something for the California Science Fair, not an article for a professional journal. @Billy: When you say the word "research," many people take that to mean "proving a theorem that no one has proven before." When other commenters say that you can't do research, they mean this sense of the word. Complex analysis is an extremely well-developed field, and proving something new often takes months of hard work. –  Jesse Madnick Jan 16 '13 at 3:33

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If you know how to program, a very fun project would be to render different kinds of fractals. The most famous, of course, is the Mandelbrot fractal, defined as the set of points $c$ in the complex plane so that if $f(z) = z^2 + c$ then the sequence $f(0), f(f(0)), f(f(f(0))), \ldots$ is bounded. It's not hard to render an image of the fractal: for each pixel on the image, take the corresponding point $c$ and color it black, if after (say) $100$ steps the sequence is still within two units of the origin. If not, you can color it various shades based on how how many steps it took to leave this disc.

You've probably seen pictures of the Mandelbrot set, but what's surprising is how simple the algorithm is. You can make amazing images with only twenty or so lines of code. You can mix it up by changing the coloring scheme or zooming in on different parts. By taking images at successive zoom levels, you can make a nice animation. You can also try replacing $f(z) = z^2 +c$ by other functions like $f(z) = z^3 + c$ or $f(z) = e^z + c^2$ or whatever you want - most of the time, you will get a very striking, surprising image. Then you can try to explain some basic features of the picture using mathematics.

Other nice do-it-yourself fractals are the Julia sets, Buddhabrot, Newton fractals, the burning ship fractal. If you don't know how to program it from scratch, there should be software that can help you (although I haven't tried these.)

Another fun thing to try is video feedback, although it's not as directly related to complex analysis. You can rent a video camera, display the camera's footage to a screen, and then point the camera at the screen so that it records itself. If you tilt the camera at different angles you can get very interesting results. See this tutorial on Youtube.

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Thanks for the clarification.

If you're up for a second, much deeper look at complex analysis, you will find that there are a great many beautiful things that can be done with the theory. My suggestion is to pick up a book that treats complex analysis rigorously and explore the topics therein. Your book already seems to address many of the applications of complex analysis (fractals, applications in celestial mechanics, etc.). Other books will address even more topics: for example, Complex Analysis by Stein and Shakarchi addresses the Riemann zeta function and the prime number theorem, both worthwhile topics for an expository paper.

If you are confident in your abilities, then there may be research topics that are accessible to you involving complex analysis: I believe there is an REU at Cornell that sometimes offers analysis on fractals as a topic, though I don't know many details about that. Perhaps a professor at your local college may be willing to advise you on the topic and tell you what you need to know to proceed in that direction.

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Here is a list of references in complex analysis: math.stackexchange.com/questions/30749/…. If you want to seriously study complex analysis consider using some of these as textbooks. –  kigen Jan 16 '13 at 3:12
    
I'm very interested in fractals, but I don't know where to begin from there. –  BillyKendrick Jan 16 '13 at 3:18
    
A professor is probably your best bet then, or maybe this deserves a new, more specifically targeted reference-request question. –  kigen Jan 16 '13 at 3:21
    
Are you saying that I start a new thread about fractals in complex analysis? –  BillyKendrick Jan 16 '13 at 3:27
    
Up to you. If you want a quick response from more knowledgeable folks in that field then this might be a good start. But take the suggestion (which many others here have made) about consulting a local mathematics professor seriously. And don't expect whatever you get to lead to a project you can present within 11 days - that's unreasonable. –  kigen Jan 16 '13 at 3:34

If you want to do something that isn't really rigourous maths for a state fair how about comparing how soap bubbles form on a closed wire to the fact that holomorphic functions on a disk are determined by their behaviour on the boundary of the disk?

Shape of wire (boundary) --- determines soap bubble on the interior

Value of Holomorphic Function on boundary --- determines the value of function on interior

Useful links: http://en.wikipedia.org/wiki/Minimal_surface, http://math.berkeley.edu/~sethian/2006/Applications/MinimalSurfaces/minimal.html, http://en.wikipedia.org/wiki/Cauchys_integral_formula, http://en.wikipedia.org/wiki/Identity_theorem,

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I'm not sure if I entirely understand how the two relate. What do you mean by closed wire? –  BillyKendrick Jan 16 '13 at 3:29
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@Billy: Imagine you have a metal wire that is shaped like a circle or ellipse or some other closed curve. If you dunk it in soapy water, then pull it out, you'll get a pretty-looking soap bubble surface (with the wire as the boundary). This can be modeled as a "minimal surface." Minimal surface theory is a topic of active research, and has connections to complex analysis. –  Jesse Madnick Jan 16 '13 at 3:38
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@Billy: It seems to me that JP is suggesting a topic in complex analysis that you might focus on (because you'll have to narrow down the scope from all of complex analysis). There could be many projects on this topic. JP is suggesting a project whereby you compare the fact that [the shape of soap bubbles are determined by the shape of their boundary wire] to the mathematical fact that [holomorphic functions on a region are determined by their values on the boundary]. For instance, you might present examples of both phenomena. –  Jesse Madnick Jan 16 '13 at 5:52
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That these two phenomena are at all related is evidenced by the relationship between minimal surfaces (soap bubbles) and holomorphic functions. @JpMcCarthy: Is this basically right? (It's been a while since I've thought about minimal surfaces...) (For the record, though, I think that modeling the shape of a soap bubble -- if you could do it -- would be a pretty cool topic.) –  Jesse Madnick Jan 16 '13 at 5:54
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@FeralOink There are links but in reality the soap bubble only illustrates the idea of the Cauchy Integral Formula. Blow into the bubble and the boundary remains the same but you have a different minimal surface. So the answer is no (although nothing should be taken from Jesse's comments either). Note that my answer started with "If you want to do something that isn't really rigourous maths..." –  Jp McCarthy Mar 26 '13 at 14:06

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