I am looking to self-study complex analysis. While I have a good grasp of undergraduate real analysis (sequences, series, analysis of single-variable functions, point-set topology), I have forgotten most of the complex analysis I have studied. The book we used was by Bak and Newman, which I didn't like much. Are there any good books or online resources available?
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I'm out of college, and trying to self-learn complex analysis. I'm finding  text difficult. Any recommendations? I'm probably at an intermediate sophistication level for an undergrad. (Bonus points if the text has a section on the Riemann Zeta function and the Prime Number Theorem.)
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I was wondering if anyone could recommend some books for studying topics such as abstract manifolds, differential forms on manifolds, integration of differential forms, Stokes' theorem, de Rham cohomology, Hodge star operator? Our text is A Comprehensive Introduction to Differential Geometry by Spivak, but I think this book is very difficult for a beginner to learn.  Thanks in advance.
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I've seen a lot of excellent visual material (gifs, pictures) here, in topics , and I used many of them to understand/explain concepts (particularly gifs showing Riemann sums or fourier series), and now I'm looking for material of this kind for the Riemann-Stieltjes integral.   Any kind would be appreciated.
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Due to my ignorance, I find that most of the references for mathematical analysis (real analysis or advanced calculus) I have read do not talk much about the "multivariate calculus". After dealing with the single variable calculus theoretically, it usually directly goes to the topic of measure theory.    After reading the wiki article "", I'd like to find the rigorous proof for this test. The first book comes to my mind is Courant's Introduction to Calculus and Analysis which includes the multivariate case in the second volume.   Motivated by this, I'd like to put the question here:     What are the usual references for the   theoretical treatment for   ?
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Dear Academic SE advisers,  I am a college sophomore in US with double majors in mathematics and microbiology. My algorithmic biology research got me passionate about the number theory and analysis, and I have been pursuing a mathematics major starting on this Spring semester. I have been independently self-studying the number theory textbooks written by Niven/Zuckerman/Montgomery, Apostol, and Ireland/Rosen on this semester. As this semester progressed, I discovered that I am more interested in the pure mathematics than applied aspects (computational biology, cryptography, etc.). I want to pursue a career as analytic number theorist and prove the Collatz conjecture and Erdos-Straus conjecture.   I have been thinking about doing the number-theory research on my university (research university; huge mathematics department). I have been self-studying the NT by myself and also regularly attending the professional and graduate seminars on number theory but I did not do any pure mathematics research as an undergraduate. Should I visit NT professors in my university and ask them about if I can do undergraduate research under them? If research is not possible (perhaps due to my lacking maturity), should I request of doing independent reading under them and later proceed with the research? How should I ask them? What should I address? If even independent reading is not desirable to them, what should I ask to them or do in my own?   As for my mathematical background, I have been taking Calculus II (computational) and discrete mathematics. I will be taking calculus III (vector calc.) on Summer, followed by Analysis I, Probability, Theoretical Linear Algebra on Fall 2015. As for my self-studying on this semester, I have been studying NT textbooks (mentioned above), proof methodologies, and basic linear algebra.   Thank you very much for your time, and I look forward to your advice!  Sincerely,  PK
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During this summer break, I want to use my time to broaden my knowledge of mathematics. I am a very motivated high school student searching for some challenging material to learn, and the two subjects I had in mind are complex analysis and probability theory.  For probability theory I already choose a book, but I am struggling a bit with finding the right complex analysis textbook. I already found two textbooks which seem pretty good to me:  by Sara Maad Sasane and  by Zill.  As to requirements for the book:   The textbook must not be too broad, since I only have 6 weeks to study intensively (yes, I know it's a challenge to study most of complex analysis in 6 weeks). The textbook must not require a lot of prerequisities other than knowledge of single variable calculus (not a course in proof writing) and must not be too rigourous (some pictures and intuitive explanations can come in handy).   I hope someone could give some advise as to if the books I found suit my needs, if someone has experience with those books or if someone knows a book other than the two I found which suit my needs better.  Thanks in advance.
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Is there any advantage to studying  single variable complex analysis as it is right now instead of just studying the world of the functions $f:\Bbb R^2 \to \Bbb R^2$?  I'm asking this because any function $\gamma: \Bbb C\to \Bbb C$ can be written as $\gamma(z)=u(x,y)+v(x,y)i$ and so we can associate the function $F: \Bbb R^2 \to \Bbb R^2, F(x,y)=(u(x,y),v(x,y))$ and study all the properties of a complex valued function as a real vector valued one.
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I have no idea if this question is appropriate for this forum, but I hope you guys can overlook the fact that I asked it on a wrong forum (if I did) and still help me answer it (of course, if this is indeed the wrong forum for the type of question I'm about to ask, do please say so).  I am a 16 year old guy who is passionate about physics and as a result wants to increase his knowledge in mathematics, the language of physics. I've read and heard a lot about Linear Algebra and how crucial it is to physics and I am deeply motivated to self study this intriguing part of mathematics. However, I have limited knowledge of maths. I (for example) know basic algebra, trig, diff/int calc and some analytical geometry, but I wouldn't say I master these subjects past the high school curriculum. Now my question is: Would you kind people say I am able to self study Linear algebra or is it too tough and/or does it require too much of a math background? And are there any good books out there for BEGINNERS in LA? I found this (free) e-book called: Elementary Linear Algebra by Kenneth Kuttler and another one called 'Linear Algebra: Theory and its applications' also by Kuttler? Are these any good? Or would you recommend other books? If you guys have any tips regarding books for LA but also tips in general, please, I'd appreciate them!
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I'm starting to read Baby Rudin (Principles of mathematical analysis) now and I wonder whether you know of any companions to it. Another supplementary book would do too. I tried Silvia's notes, but I found them a bit too "logical" so to say. Are they good? What else do you recommend?
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Though there are several posts discussing the reference books for topology, for example . But as far as I looked up to, all of them are for the purpose of learning topology or rather on introductory level.  I am wondering if there is a book or a set of books on topology like Rudin's analysis books, S.Lang's algebra, Halmos' measure (or to be more updated, Bogachev's measure theory), etc, serving as standard references.  Probably, such a book should hold characteristics such as being self-contained, covering the most of classical results, and other good properties you can name.  In the end, I only have the interest on general-topology (topological space, metrization, compactification...), and optionally differential topology (manifolds). So please don't divert into algebraic context.  Cheers.  ---------------------update-----------------------  Thanks for all the delightful replies. I will try to quickly check the books mentioned beneath. And I may accept the answer which is most closed to my personal flavor.  Sorry for others. It's a pity no multiple acceptance can be made for such a ref-request question.  Greetings.
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Since summer comes with a lot of spare time, I've decided to select a mathematical subject I want to learn as much as possible about over the next three months. That being said, number theory really caught my eye, but I have no prior training in it.   I've decided to conduct my studying effort in a library; I prefer real books to virtual ones, but as I'm not allowed to browse through the books on my own, I have to know beforehand what I'm looking for and this is where I'm kind of lost. I'm not really sure where to start.  Basically I wanna know about the following:    What are the prerequisites? (I'm currently trained in Linear Algebra, Calculus, Complex Analysis - all on an undergraduate level )  Can you recommend some reading materials?     Thank you.
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Edit:  Ok, user  made comment that truly helps narrow the focus of my question. Basically, I'm asking for a self-study course of Mathematical Methods.  Thanks to his recommendation I identified a book that is very close to the answer:  Mathematical Methods in the Physical Sciences by Mary L. Boas  Supposedly, the book covers most of the Math a person needs to learn in order to understand the simulations I presented.  Does this book suit my needs?    Hello,  I have always been interested on the idea to study Math at an Undergraduate level to be able to implement simulations like the followings:  Finite Difference Simulation of the Wave Equation    Heat Equation in Two Dimensions    Besides my precollege experience in Math, I have only been able to do very simple simulations due to my lack of knowledge of Advanced Math. Every time I ask about the inner workings of the simulations I get redirected to papers that look like foreign language to me and, unfortunately, I depend on easy-to-read tutorials to understand them, and while I have found   I'm more interested in reaching the solution myself.  I checked some of the questions related to book recommendations for self-study and while they are very helpful I'm interested in learning by example (There is a book called: "Accelerated C++ Practical Programming by Example" that fits very well my study habits)  Thanks in advance.
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What is your recommendation for an in-depth introductory combinatoric book? A book that doesn't just tell you about the multiplication principle, but rather shows the whole logic behind the questions with full proofs. The book should be for a first-year-student in college. Do you know a good book on the subject?  Thanks.
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I am studying Apostol's book of Mathematical Analysis, and I am looking for problems. I have solved most of the problems in Apostol and most in Rudin, also from Folland. Can someone plese recommend any other book with decent problems? I know this question has been asked before. I need mostly proof problems. Any help or reference would be appreciated.  Thank in advance!
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I'm looking for an understandable proof of this theorem, and also a complex one involving beautiful math techniques such as analytic number theory, or something else. I hope you can help me on that. Thank you very much
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The question is #$14$ from Chapter $2$ in Stein and Shakarchi's text Complex Analysis:     Suppose that $f$ is holomorphic in an open set containing the closed unit disc, except for a pole at $z_0$ on the unit circle. Show that if $$\sum_{n=0}^\infty a_nz^n$$ denotes the power series expansion $f$ in the open unit disc, then $$\lim_{n\to\infty}\frac{a_n}{a_{n+1}}=z_0.$$   I've shown that we can take $z_0=1$ without a loss of generality, but I'm having trouble showing the proof otherwise. One of the problems I'm having is because we aren't told the definition of a pole except that it is a place where the function isn't holomorphic. Disregarding this fact, the other problem I'm running into is that I don't know the order of the pole.  Making some additional assumptions, including that the pole is simple so we can write $F(z)=(z-1)f(z)$ as a holomorphic function, we see that $$F(z)=-a_0+z(a_0-a_1)+z^2(a_1-a_2)+\cdots$$ This almost gets me to the end with these added assumptions, but I don't think it's quite enough (why do we know some of the $a_i$'s aren't $0$, for example).  On another note, if we know $\lim_{n\to\infty}\frac{a_n}{a_{n+1}}$ exists, then it is easy to see that $\lim_{n\to\infty}\frac{|a_n|}{|a_{n+1}|}=1=|z_0|$; I, however, do not see why the limit must exist.  Are there any hints that someone can provide? Even a solution would be nice, especially if one can avoid making any assumptions about what a pole is or is not.  EDIT: So there isn't any confusion, I know the definition of a pole and I'm inclined to believe that the problem, as stated, necessarily has a pole at $z_0$. The problem is that the exercise is in Chapter $2$, and poles are introduced in Chapter $3$.
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I was studying some hyperbolic geometry previously and realised that I needed to understand things in a more general setting in terms of a "manifold" which I don't yet know of.  I was wondering if someone can recommend to me some introductory texts on manifolds, suitable for those that have some background on analysis and several variable calculus. A lecturer recommended to me "Analysis on Real and Complex Manifolds" by R. Narasimhan, but it is too advanced.   I had a look at Loring W. Tu's text on manifolds and it seemed accessible.