This tag is for questions where the poster seeks references (books, articles, etc.) about a particular subject. Please do not use this as the only tag for a question.

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33
votes
5answers
4k views

Completion of rational numbers via Cauchy sequences

Can anyone recommend a good self-contained reference for completion of rationals to get reals using Cauchy sequences?
22
votes
14answers
13k views

best book for topology?

I am a graduate student of math right now but i was not able to get a topology subject in my undergrad... i just would like to know if you guys know the best one..
7
votes
1answer
776 views

half-derivative of $x^2$

I was given this problem to challenge me $$\frac{d^{1/2}}{dx^{1/2}}x^2 $$ I googled wikipedia, and tried to follow the steps shown. I got an answer of $\frac{16\sqrt{ \pi x}}{9\pi}$ edited 2 ...
70
votes
27answers
21k views

Best book ever on Number Theory

Which is the single best book for Number Theory that everyone who loves Mathematics should read?
40
votes
23answers
12k views

What is a good complex analysis textbook?

I'm out of college, and trying to learn complex analysis on my own. I took out Ahlfors' text from the library, but I'm finding it difficult. Any textbook recommendations? I'm probably at an ...
20
votes
3answers
1k views

Every power series is the Taylor series of some $C^{\infty}$ function

Do you have some reference to a proof of the so-called Borel theorem, i.e. every power series is the Taylor series of some $C^{\infty}$ function?
48
votes
8answers
11k views

Teaching myself differential topology and differential geometry

I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology. I have decided to fix this lacuna once for ...
17
votes
5answers
3k views

What is a good book to study linear algebra?

I'm looking for a book to learn Algebra. The programme is the following. The units marked with a $\star$ are the ones I'm most interested in (in the sense I know nothing about) and those with a ...
8
votes
4answers
3k views

Undergraduate/High-School-Olympiad Level Introductory Number Theory Books For Self-Learning

I don't know whether the books metioned in Best ever book on Number Theory are beyond undergraduate/high-school-olympiad level. Please recommend your favourite.
12
votes
7answers
2k views

Choosing a text for a First Course in Topology

Which is a better textbook - Dugundji or Munkres? I'm concerned with clarity of exposition and explanation of motivation, etc.
2
votes
5answers
449 views

Does $i^i$ and $i^{1\over e}$ have more than one root in $[0, 2 \pi]$

How to find all roots if power contains imaginary or irrational power of complex number? How do I find all roots of the following complex numbers? $$(1 + i)^i, (1 + i)^e, (1 + i)^{ i\over e}$$ EDIT:: ...
22
votes
7answers
21k views

What is the best book for studying discrete mathematics?

As a programmer, mathematics is important basic knowledge to study some topics, especially Algorithms. Many websites, and my fellows suggest me to study Discrete Mathematics before going to ...
42
votes
2answers
979 views

Reference request for tricky problem in elementary group theory

The following could have shown up as an exercise in a basic Abstract Algebra text, and if anyone can give me a reference, I will be most grateful. Consider a set $X$ with an associative law of ...
21
votes
10answers
2k views

Seeking a layman's guide to Measure Theory

I would like to teach myself measure theory. Unfortunately most of the books that I've come across are very difficult and are quick to get into Lemmas and proofs. Can someone please recommend a ...
8
votes
8answers
5k views

What are the recommended textbooks for introductory calculus?

I've already taken my calculus sequence and I'm interested in brushing up and staying sharp on the basics. So far, my calculus background is limited to single-variable calculus, which I applied in my ...
13
votes
11answers
3k views

Requesting abstract algebra book recommendations

I've taken up self-study of math. (How smart can that be?) I've just about finished a course in real analysis which spent a lot of time on metric spaces and some time revisiting calculus. I was ...
11
votes
2answers
680 views

Infinite products - reference needed!

I am looking for a small treatment of basic theorems about infinite products ; surprisingly enough they are nowhere to be found after googling a little. The reason for this is that I am beginning to ...
4
votes
1answer
758 views

Mean distance from origin after $N$ equal steps of Random-Walk in a $d$-dimensional space.

I am looking for a formula that evaluates the mean distance from origin after $N$ equal steps of Random-Walk in a $d$-dimensional space. Such a formula was given by "Henry" to a question by "Diego" ...
32
votes
10answers
6k views

Best Algebraic Geometry text book? (other than Hartshorne)

Lifted from Mathoverflow: I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best. Then what might be the 2nd best? It can be a book, preprint, online lecture note, ...
29
votes
5answers
4k views

(Theoretical) Multivariable Calculus Textbooks [duplicate]

(Note that I have used bold text frequently simply to highlight the key points of my question for those who do not have the time to read through it thoroughly (it is not very long, however); I hope ...
14
votes
4answers
523 views

Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$

I was reviewing some matrices and found this interesting if $r = \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}$ then $rr=-I$, also $$\exp{(\theta r)} = \cos\theta I + \sin\theta r$$ No wonder, the ...
26
votes
10answers
2k views

Online resources for learning Mathematics

Not sure if this is the place for it, but there are similar posts for podcasts and blogs, so I'll post this one. I'd be interested in seeing a list of online resources for mathematics learning. As ...
31
votes
14answers
8k views

Good book for self study of functional analysis

I am a EE grad. student who has had one undergraduate course in real analysis (which was pretty much the only pure math course that I have ever done). I would like to do a self study of some basic ...
25
votes
13answers
11k views

Good book for self study of a First Course in Real Analysis

Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used "Introduction ...
10
votes
8answers
743 views

Book about technical and academic writing

I'm in the process of writing my Master's Thesis on automata theory. The writing must be in English which isn't my mother tongue. So the question is, given that this is my first time long (hundred ...
6
votes
6answers
6k views

Good First Course in real analysis book for self study

Does anybody know of a good book in real analysis for self study for a beginner? What about Analysis 1 by Terence Tao?
4
votes
4answers
2k views

The characteristic and minimal polynomial of a companion matrix

The companion matrix of a monic polynomial in 1 variable over a field $F$ plays an important role in understanding the structure of finite dimensional $F[x]$-modules. It is an important fact that the ...
47
votes
17answers
9k views
18
votes
6answers
11k views

Good books for self-studying algebra?

I have a few weeks off from school soon, and I was hoping to self-study a bit of algebra. I don't think this question has been asked on here before, but does anyone have any suggestions for algebra ...
55
votes
5answers
2k views

Defining a manifold without reference to the reals

The standard definition I've seen for a manifold is basically that it's something that's locally the same as $\mathbb{R}^n$, without the metric structure normally associated with $\mathbb{R}^n$. ...
14
votes
8answers
3k views

References for the multivariate calculus

Maybe 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 ...
12
votes
6answers
6k views

Book recommendation on plane Euclidean geometry

I consider myself relatively good at math, though I don't know it at a high level (yet). One of my problems is that I'm not very comfortable with geometry, unlike algebra, or to restate, I'm much more ...
16
votes
6answers
12k views

Good abstract algebra books for self study

Last semester I picked up an algebra course at my university, which unfortunately was scheduled during my exams of my major (I'm a computer science major). So I had to self study the material, ...
7
votes
12answers
4k views

Introductory Group theory textbook

What textbook is good for introductory group theory?
6
votes
5answers
570 views

What are or where can I find style guidelines for writing math?

I am a scientist writing my first manuscript with a substantial amount of mathematical methodological documentation. I am using LaTeX, but this is not my question. I would like to find a list of ...
19
votes
9answers
3k views

Reference book on measure theory

I post this question with some personal specifications. I hope it does not overlap with old posted questions. Recently I strongly feel that I have to review the knowledge of measure theory for the ...
15
votes
2answers
322 views

How can I prove my conjecture for the coefficients in $t(x)=\log(1+\exp(x)) $?

I'm considering the transfer-function $$ t(x) = \log(1 + \exp(x)) $$ and find the beginning of the power series (simply using Pari/GP) as $$ t(x) = \log(2) + 1/2 x + 1/8 x^2 – 1/192 x^4 + 1/2880 x^6 - ...
6
votes
1answer
1k views

Basic facts about ultrafilters and convergence of a sequence along an ultrafilter

Could you help, please. I need the information about the ultrafilters, namely, any ideas how one can see that they exist and a proof of the fact that for any ultrafilter every sequence on a compact ...
6
votes
3answers
531 views

Looking for a Calculus Textbook

I want to start signal processing and I need a book that satisfies my mathematical requirements: I am in the third grade of high school and I don't know any useful thing about limit, differential, ... ...
5
votes
2answers
210 views

Solving simple congruences by hand

When I am faced with a simple linear congruence such as $$9x \equiv 7 \pmod{13}$$ and I am working without any calculating aid handy, I tend to do something like the following: "Notice" that adding ...
9
votes
1answer
561 views

Constructing a subset not in $\mathcal{B}(\mathbb{R})$ explicitly

While reading David Williams's "Probability with Martingales", the following statement caught my fancy: Every subset of $\mathbb{R}$ which we meet in everyday use is an element of Borel ...
4
votes
2answers
763 views

Introductory book on Topology [duplicate]

Possible Duplicate: choosing a topology text best book for topology? What book would you recommend for an undergraduate wanting to learn the basics of topology? I've come across several ...
66
votes
4answers
2k views

Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$

I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$. I found a solution by myself 10 hours after I posted it, here it is: ...
21
votes
3answers
1k views

Consequences of Degree Theory

I'm preparing a presentation on an overview of algebraic and differential topology, and my introduction includes some motivational material on Degree Theory. I have two fundamental and invaluable ...
22
votes
4answers
3k views

Learning Roadmap for Algebraic Topology

I am now considering about studying algebraic topology. There are a lot of books about it, and I want to choose the most comprehensive book among them. I have a solid background in Abstract Algebra, ...
20
votes
9answers
1k views

Where to begin with foundations of mathematics

I would like to know more about the foundations of mathematics, but I can't really figure out where it all starts. If I look in a book on axiomatic set theory, then it seems to be assumed that one ...
16
votes
6answers
5k views

What are good books/other readings for elementary set theory?

I am looking to expand my knowledge on set theory (which is pretty poor right now -- basic understanding of sets, power sets, and different (infinite) cardinalities). Are there any books that come to ...
32
votes
2answers
1k views

Is there an explicit isomorphism between $L^\infty[0,1]$ and $\ell^\infty$?

Is there an explicit isomorphism between $L^\infty[0,1]$ and $\ell^\infty$? In some sense, this is a follow-up to my answer to this question where the non-isomorphism between the spaces $L^r$ ...
16
votes
8answers
4k views

What are good books to learn graph theory?

What are some of the best books on graph theory, particularly directed towards an upper division undergraduate student who has taken most the standard undergraduate courses? I'm learning graph theory ...
17
votes
4answers
947 views

probability textbooks

Has anyone compiled a moderately comprehensive list on the web or elsewhere of textbooks on probability For students who have not been introduced to the subject before That introduce both discrete ...