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

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35
votes
16answers
27k 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..
9
votes
3answers
563 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 ...
47
votes
5answers
6k 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?
29
votes
6answers
12k 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 ...
110
votes
29answers
38k views

Best book ever on Number Theory

Which is the single best book for Number Theory that everyone who loves Mathematics should read?
28
votes
3answers
2k 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?
65
votes
23answers
26k 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 ...
8
votes
1answer
1k 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 ...
79
votes
9answers
21k 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 ...
53
votes
17answers
30k 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 ...
75
votes
22answers
18k views
26
votes
4answers
41k views

Eigenvectors of real symmetric matrices are orthogonal

Can someone point me to a paper, or show here, why symmetric matrices have orthogonal eigenvectors? In particular, I'd like to see proof that for a symmetric matrix $A$ there exists decomposition $A = ...
15
votes
12answers
8k 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 ...
40
votes
5answers
9k 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 ...
22
votes
7answers
24k 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, ...
12
votes
4answers
4k views

Overview of basic results about images and preimages

Are there some good overviews of basic facts about images and inverse images of sets under functions?
7
votes
1answer
1k views

Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent.

This is solely a reference request. I have heard a few versions of the following theorem: If the joint moment generating function $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ ...
53
votes
2answers
1k 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 ...
26
votes
12answers
6k 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 ...
14
votes
7answers
4k 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.
13
votes
2answers
1k 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 ...
15
votes
4answers
747 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 ...
8
votes
5answers
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 ...
3
votes
3answers
4k views

an example of a continuous function whose Fourier series diverges at a dense set of points

Please give me a link to a reference for an example of a continuous function whose Fourier series diverges at a dense set of points. (given by Du Bois-Reymond). I couldn't find this in Wikipedia.
5
votes
3answers
896 views

Erdös-Straus conjecture

I'm reading a lot about the Erdös-Straus Conjecture (ESC), a conjecture that states that for every natural number $p \geq 2$, there exists a set of natural numbers $a, b, c$ , such that the following ...
40
votes
8answers
40k 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 ...
29
votes
10answers
4k 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 ...
19
votes
9answers
36k views

Calculus book recommendations (for complete beginner)

Well I have not started calculus yet but I am really keen to. I would love if you suggest some books. Points to be noted: I really don't like the way textbooks are written so please no "textbooks" ...
13
votes
5answers
11k views

Best Algebraic Topology book/Alternative to Allen Hatcher free book?

Allen Hatcher seems impossible and this is set as the course text? So was wondering is there a better book than this? It's pretty cheap book compared to other books on amazon and is free online. ...
10
votes
4answers
4k 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.
2
votes
5answers
561 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:: ...
46
votes
10answers
10k 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, ...
34
votes
6answers
28k 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 ...
42
votes
15answers
15k 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 ...
28
votes
9answers
2k 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 ...
20
votes
6answers
9k 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 ...
12
votes
7answers
8k 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 ...
15
votes
1answer
612 views

Alternative construction of the tensor product (or: pass this secret)

The paper Tensor products and bimorphisms by B. Banachewski and E. Nelson studies tensor products (defined by classifying bimorphisms) in concrete categories. It is quite interesting that their main ...
6
votes
1answer
538 views

Treatise on non-elementary integrable functions

All of us mathematicians after some time (and trial-and-error, of course) we are able to guess with reasonable accuracy whether or not a given function is elementary integrable (test yourself: ...
3
votes
1answer
517 views

Characterization of integers which has a $2$-adic square root

Does anyone know an "elementary" proof of the following theorem? Let $k \neq 0$ be a rational integer. Then $k$ admits a square root in $\mathbb{Z}_2$ if $k = 4^a (8b+1)$ for some $a \in \mathbb{N}$, ...
9
votes
1answer
946 views

Reference book for Artin-Schreier Theory

The aim of the question is very simple, I would like to study Artin-Schreier Theory, but I have had embarassing difficulties in finding a book which could help me in doing that. In specific I'm ...
5
votes
1answer
1k 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" ...
158
votes
4answers
12k views

The Integral that Stumped Feynman?

In "Surely You're Joking, Mr. Feynman!," Nobel-prize winning Physicist Richard Feynman said that he challenged his colleagues to give him an integral that they could evaluate with only complex methods ...
19
votes
1answer
3k views

Overview of basic results on cardinal arithmetic

Are there some good overviews of basic formulas about addition, multiplication and exponentiation of cardinals (preferably available online)?
11
votes
8answers
1k 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 ...
23
votes
9answers
2k views

Very good linear algebra book.

I plan to self-study linear algebra this summer. I am sorta already familiar with vectors, vector spaces and subspaces and I am really interested in everything about matrices (diagonalization, ...), ...
17
votes
3answers
2k views

Squares in arithmetic progression

It is easy to find 3 squares (of integers) in arithmetic progression. For example, $1^2,5^2,7^2$. I've been told Fermat proved that there are no progressions of length 4 in the squares. Do you know ...
13
votes
2answers
2k views

Every partial order can be extended to a linear ordering

How do I show that every partial order can be extended to a linear ordering? I think that I manage to prove that claim for finite set, how can I prove it for infinite set? Thank you.
6
votes
1answer
2k 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 ...
12
votes
1answer
1k 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 ...