Complex Analysis in High school

[hddnvariables]

I am currently a freshman in high school in the United States. I am very interested in mathematical and theoretical physics. For about the past year or two I have been studying on my own.
Over the last summer I spent most of July and August reading Calculus by James Stewart and feel that I have a firm understanding of the information presented in the text. I have also read several other books and am currently working on An introduction to Quantum Mechanics with Applications to Chemistry by Linus Pauling and E. Bright Willson Jr., as well as The Road to Reality by Roger Penrose. Last year my high school physics teacher gave me Physics by Giancoli to study from, and I am currently taking Physics concurrently with AP Physics B, the highest physics course offered at my school. In math I am taking an independent study of Algebra 2 for the first semester and an online Advanced Trig. course the second semester.
I understand that many of the topics covered in these courses are important in higher mathematics, but I do not believe that I need this long to learn them. If I follow my current school schedule I will be taking AP Calculus BC, the most advanced mathematics course at my high school, my senior year. I have already learned most of the information covered in this section and would only need a brief refresher to get a firm grasp on the subject. In any spare time I can find outside of school I have continued to study independently. Currently I am learning about Complex Analysis and related mathematical concepts. I have looked into it and found that a few boarding schools offer classes that are this advanced, but I am not enthusiastic about attending one of these. My school is a small public school and has, in the past, offered some support to students that are excelling in their classes. I have been wondering if there was any way that I could take these advanced courses, such as Complex Analysis or possibly Quantum Mechanics, in high school. I have been told that I can take AP tests without actually taking the AP courses, but don't want to risk missing some important concept that is not covered in the books that I read and I am not sure how this looks on a college application.
I have also been wondering if there were any specific books that anyone would recommend for someone who is trying to learn through an independent study. Currently I am scavenging most of my books through resale shops. I found one store that has a large selection of Dover books and was wondering if these would be suitable for my studies. Is there any place I can purchase good textbooks for low prices?
I would really appreciate any advice that anyone could give me. I am open to any advice or criticism. Thanks for any advice.

 

[lurflurf]

If there is a college near by it might be possible to take a few courses there while in high school. It is good to be able to learn things sometimes without talking a class. I think most college students take a least one class that is worse than nothing.

I suggest reading this book
Elementary Real and Complex Analysis (Dover Books on Mathematics) by Georgi E. Shilov
It will shore up your understanding of calculus (if needed) and provide a brief introduction to complex analysis.

 

[SpaceDreamer]

How did you study and grasp calculus before taking algebra 2 and trig, they provide essential information needed to understand calculus. Also you can get cheap textbooks by older older ones even online, once a semester is past textbook prices will start to drop like a rock as new additions are released.

 

[micromass]

What makes you think you can handle complex analysis? Complex analysis requires quite some prerequisite knowledge such as things from real analysis and topology. Furthermore, you need to be extremely proficient in proofs and you should have the mathematical maturity required to read an analysis book.
As of now, you haven't read an actual math book yet, so somehow I doubt you are ready for complex analysis.

I suggest you pick up a copy of "Calculus" by Spivak. Don't be mislead by the title: the book is more of an intro-analysis book than an actual calculus book. The text and the problem are very challenging! Furthermore, Spivak actually contains some chapters on complex analysis in the end of the book: specifically, he proofs the fundamental theorem of algebra.
I highly suggest you try to work through Spivak first.

 

[espen180]

I agree with micromass. Complex Analysis is probably too advanced for you at the moment, unless you mean a complex variables-style course where you only learn methods and no proofs. You still need to master both single and multivariable calulus though. You also need to know about convergence of sequences and analytic geometry.
See for example this:
http://ocw.mit.edu/resources/res-18...quations-and-linear-algebra-fall-2011/part-i/

 

[lurflurf]

One can learn complex numbers and calculus at the same time. Physicists do, they use it and they do not know any math. In math knowing things helps to learn other things, but it is not essential.

 

[Klungo]

What maths exactly do you know?

Personally, I see that roughly several lower division math courses: calc sequence, DE, introductory algebra to be mostly computational and not actually preparing you for analysis. Let alone taking the AP version. (You could do very poorly on the BC topics on the exam and still pass with a 4-5 on the BC exam, ridiculous!)

You need to first look into writing good proofs as most of analysis is proofs. As micromass says, prerequsites of complex analysis are real analysis and topology. Although they may not strictly require a topology course, it will surely appear on the real analysis course.

Complex variables on the other hand, goes through the algebra, analyticity, functions, contour integration, residue calculus all with (or without) some theory. My instructor of this course gives out a lot of proofs but they aren't really too rigourous. Even then, this course requires multivariable calculus and "differential equations". (You work with very simple PDEs but they don't require too much skill nor does it really use ODE).

 

[halo31]

Maybe what you are looking for is more of an applied complex analysis book which I think you can take on but there will still be some rigorous proofs. The dover series are actually quite good for self study so it wouldn't hurt in trying it out. IF you find it a bit too hard just try taking a class in proofs. As a matter of fact Coursera offers one called Mathematical Thinking which is quite good and its free.

 

[hddnvariables]

Thanks for the advice. When I mentioned Complex Analysis I just meant to use it as an example of an advanced course. Much of the math section in The Road to Reality by Roger Penrose focusses on Complex Analysis. I found that the book left many gaps and over simplified many of the topics and was hoping to fill these in. Really I am just looking for any advice on how I can study ahead.
SpaceDreamer, in the book that I read, Calculus by James Stewart, I found that my background knowledge was enough to thoroughly understand the text. In Algebra Two I find that most of the information is repeated from prior courses or can be derived directly from information in prior courses. The same is true with the trigonometry in the book. Any information that I did lack was supplied throughout the book in the proofs.
Klungo, Calculus covered differentiation, integration, applications of these, sequences, partial differentiation, vector calculus, basic first and second order differential equations, and an introduction to partial differential equations. I have also studied math from several websites and other books.
My main goal is to learn as much as I can. Many of you have suggested topology and analysis. What are some good resources for these? I appreciate the advice that I have received so far and hope to receive more soon.

 

[micromass]

I should have mentioned this earlier, but this book is a real gem in the teaching of mathematical ideas: https://www.amazon.com/dp/1402021860/?tag=pfamazon01-20
It covers a bit of abstract algebra and complex analysis. The main goal of the text is to "prove" Abel's theorem that roots of quintics can not explicitely be found using the elementary operations.
The book came from a lecture series that Arnold gave to school children! So the book is perfectly suitable for (a very motivated) high school student, but the book does contain very mathematically advanced topics. For example, Riemann surfaces are a mathematical object that most math majors probably don't hear about in undergrad (I certainly didn't), but this book introduces it anyway in a way that it very intuitive and not very hard.
While the book is not immediately a very rigorous and formal text, I do think it presents the concepts extremely well.

If you're interested in real analysis and topology, then I really suggest that you read Spivak's calculus. It is the perfect introduction to proofy mathematics such as analysis. It is a very challenging book, but it is also very beautiful.