(Part 2) Reddit mentions: The best mathematical analysis books

This is a compilation of what I gathered from reading on the internet about self-learning higher maths, I haven't come close to reading all this books or watching all this lectures, still I hope it helps you.

General Stuff:
The books here deal with large parts of mathematics and are good to guide you through it all, but I recommend supplementing them with other books.

1. Mathematics: A very Short Introduction : A very good book, but also very short book about mathematics by Timothy Gowers, a Field medalist and overall awesome guy, gives you a feelling for what math is all about.
   
   
2. Concepts of Modern Mathematics: A really interesting book by Ian Stewart, it has more topics than the last book, it is also bigger though less formal than Gower's book. A gem.
   
   
3. What is Mathematics?: A classic that has aged well, it's more textbook like compared to the others, which is good because the best way to learn mathematics is by doing it. Read it.
   
   
4. An Infinitely Large Napkin: This is the most modern book in this list, it delves into a huge number of areas in mathematics and I don't think it should be read as a standalone, rather it should guide you through your studies.
   
   
5. The Princeton Companion to Mathematics: A humongous book detailing many areas of mathematics, its history and some interesting essays. Another book that should be read through your life.
   
   
6. Mathematical Discussions: Gowers taking a look at many interesting points along some mathematical fields.
   
   
7. Technion Linear Algebra Course - The first 14 lectures: Gets you wet in a few branches of maths.
   
   Linear Algebra: An extremelly versatile branch of Mathematics that can be applied to almost anything, also the first "real math" class in most universities.
   
   
8. Linear Algebra Done Right: A pretty nice book to learn from, not as computational heavy as other Linear Algebra texts.
   
   
9. Linear Algebra: A book with a rather different approach compared to LADR, if you have time it would be interesting to use both. Also it delves into more topics than LADR.
   
   
10. Calculus Vol II : Apostols' beautiful book, deals with a lot of lin algebra and complements the other 2 books by having many exercises. Also it doubles as a advanced calculus book.
    
    
11. Khan Academy: Has a nice beginning LinAlg course.
    
    
12. Technion Linear Algebra Course: A really good linear algebra course, teaches it in a marvelous mathy way, instead of the engineering-driven things you find online.
    
    
13. 3Blue1Brown's Essence of Linear Algebra: Extra material, useful to get more intuition, beautifully done.
    
    Calculus: The first mathematics course in most Colleges, deals with how functions change and has many applications, besides it's a doorway to Analysis.
    
    
14. Calculus: Tom Apostol's Calculus is a rigor-heavy book with an unorthodox order of topics and many exercises, so it is a baptism by fire. Really worth it if you have the time and energy to finish. It covers single variable and some multi-variable.
    
    
15. Calculus: Spivak's Calculus is also rigor-heavy by Calculus books standards, also worth it.
    
    
16. Calculus Vol II : Apostols' beautiful book, deals with many topics, finishing up the multivariable part, teaching a bunch of linalg and adding probability to the mix in the end.
    
    
17. MIT OCW: Many good lectures, including one course on single variable and another in multivariable calculus.
    
    Real Analysis: More formalized calculus and math in general, one of the building blocks of modern mathematics.
    
    
18. Principle of Mathematical Analysis: Rudin's classic, still used by many. Has pretty much everything you will need to dive in.
    
    
19. Analysis I and Analysis II: Two marvelous books by Terence Tao, more problem-solving oriented.
    
    
20. Harvey Mudd's Analysis lectures: Some of the few lectures on Real Analysis you can find online.
    
    Abstract Algebra: One of the most important, and in my opinion fun, subjects in mathematics. Deals with algebraic structures, which are roughly sets with operations and properties of this operations.
    
    
21. Abstract Algebra: Dummit and Foote's book, recommended by many and used in lots of courses, is pretty much an encyclopedia, containing many facts and theorems about structures.
    
    
22. Harvard's Abstract Algebra Course: A great course on Abstract Algebra that uses D&F as its textbook, really worth your time.
    
    
23. Algebra: Chapter 0: I haven't used this book yet, though from what I gathered it is both a category theory book and an Algebra book, or rather it is a very different way of teaching Algebra. Many say it's worth it, others (half-jokingly I guess?) accuse it of being abstract nonsense. Probably better used after learning from the D&F and Harvard's course.
    
    There are many other beautiful fields in math full of online resources, like Number Theory and Combinatorics, that I would like to put recommendations here, but it is quite late where I live and I learned those in weirder ways (through olympiad classes and problems), so I don't think I can help you with them, still you should do some research on this sub to get good recommendations on this topics and use the General books as guides.

The Gagliargo-Nirenberg inequalities you mention originate here.

Some of Nirenberg's "greatest hits" at a glance: some of his early work concerned the Minkowski problem of finding surfaces with prescribed Gauss curvature, and the related Weyl problem of finding isometric embeddings of positive curvature metrics on the sphere. For a gentle introduction to this type of problem accessible (in principle) after a basic course in differential geometry and some analysis, see these notes by Khazdan. For a more advanced treatment, including a discussion of the Minkowski problem and generalizations see these notes by Guan. This line of research owes a lot to Nirenberg.

In this legendary paper (2700+ citations, for a math paper!) and another with the same co-authors (Agmon and Douglis), he investigated boundary Schauder and L^p estimates for solutions of general linear elliptic equations. You can look at Gilbarg-Trudinger, or Krylov's books (1, 2) for the basics of linear elliptic equations, including boundary estimates. Here is a course by Viaclovsky in case you don't want to buy the books. This last set is far more basic stuff than Agmon-Douglis-Nirenberg, though, but it should give you an idea of what its about.

Another extremely famous contribution of Nirenberg is his introduction with Kohn of the (Kohn-Nirenberg) calculus of pseudodifferential operators. Shortly thereafter, Hoermander began his monumental study of the subject, later summarized in his books I, II, III, IV. If you know nothing about pseudo-differential operators, I suggest starting with this book by Alinhac and Gérard.

Another gigantic result is the Newlander-Nirenberg theorem on integrability of almost complex structures. An almost-complex structure is a structure on the tangent space of a manifold which mimics the effect that rotation by i has on the tangent vectors. The Newlander-Nirenberg tells you that if a certain simple necessary condition holds, you can actually choose locally holomorphic coordinates for the manifold compatible which induce this a.c. structure. A proof that should be reasonably accessible, provided you understand what I just wrote and have some basic notions of several complex variables can be found here.

Nirenberg also studied the important problem of (local) solvability of (pseudo)-differential equations with Francois Treves. In this paper, he introduced the famous condition Psi, which was only recently proved by Dencker to be necessary and sufficient for local solvability. An exposition of the problem at a basic level can be found in this undergrad thesis from UW.

Another massively influential paper was this one, with Fritz John, where he introduces the space of BMO functions, and proved the Nirenberg-John lemma to the effect that any BMO function is exponentially integrable. Fefferman later identified BMO as the dual of the Hardy space Re H_1, and the BMO class plays a crucial role in the Calderon-Zygmund theory of singular integral operators. You can read about this in any decent book on harmonic analysis. I myself like Duoandicoetxea's Fourier Analysis. BMO functions are treated in chapter 6. For a more "old school" treatment using complex analysis, including a proof of Fefferman's theorem, check out Koosis' lovely Introduction to H^p spaces.

Another noted contribution was his "abstract Cauchy-Kowalevski" theorem, where he formulated the classical theorem in terms of an iteration in a scale of spaces, instead of the more direct treatment based on power series. This point of view has now become classical. Look at the proof in Treve's book Basic Linear Partial Differential Equations.

Next, his landmark paper with Gidas and Ni (2000+ citations) on symmetry of positive solutions to second order nonlinear elliptic PDE are absolute classics. The technique is now a basic part of the "elliptic toolbox".

His series of papers with Caffarelli, Spruck and Kohn (starting here) on fully nonlinear equations is also classic, and the basis for much of the later work. It's gotten sustained attention in part because optimal transport equations are of (real) Monge-Ampere type.

The theorem about partial regularity of NS you are referring to is this absolute classic with Cafarelli and Kohn. A simple recent proof, together with an accessible exposition of de Georgi's method, can be found here.

Let me finish by mentioning my personal favorite, one of the most cited papers in analysis of the 20th century, an absolute landmark of variational analysis, Brezis-Nirenberg 1983. A pedagogical exposition appears in Chapter III of Struwe's excellent book.

TLDR: Nirenberg is one of the most important analysts of the past 60 years.

edit: Thanks for the gold! Glad this was useful/interesting to someone, given how advanced and specialized the material is.

Pick up mathematics. Now if you have never done math past the high school and are an "average person" you probably cringed.

Math (an "actual kind") is nothing like the kind of shit you've seen back in grade school. To break into this incredible world all you need is to know math at the level of, say, 6th grade.

Intro to Math:

1. Book of Proof by Richard Hammack. This free book will show/teach you how mathematicians think. There are other such books out there. For example,
   
   

• How to Prove It: A Structured Approach by Daniel Velleman
  
  
• Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand et al
  
  
• Discrete Mathematics with Applications by Susanna Epp
  
  These books only serve as samplers because they don't even begin to scratch the surface of math. After you familiarized yourself with the basics of writing proofs you can get started with intro to the largest subsets of math like:
  
  Intro to Abstract Algebra:
  
  
• Linear Algebra: Step by Step by Singh
  
  
• Abstract Algebra: A Student-Friendly Approach by the Dos Reis
  
  
• Numbers and Symmetry: An Introduction to Algebra by Johnston/Richman
  
  
• Discovering Group Theory: A Transition to Advanced Mathematics by Barnard/Neil
  
  
• A Friendly Introduction to Group Theory by Nash
  
  
• Linear Algebra Done Right by Sheldon Axler
  
  There are tons more books on abstract/modern algebra. Just search them on Amazon. Some of the famous, but less accessible ones are
  
  
• Algebra: Chapter 0 by Aluffi
  
  
• Algebra by Maclane/Birkhoff
  
  
• Algebra by Lang
  
  
• Advanced Linear Algebra by Steven Roman
  
  Intro to Real Analysis:
  
  
• The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Grinberg
  
  
• Understanding Analysis by Abbot
  
  
• Writing Proofs in Analysis by Kane
  
  
• The Lebesgue Integral for Undergraduates by Johnston
  
  Again, there are tons of more famous and less accessible books on this subject. There are books by Rudin, Royden, Kolmogorov etc.
  
  Ideally, after this you would follow it up with a nice course on rigorous multivariable calculus. Easiest and most approachable and totally doable one at this point is
  
  
• Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach by the Hubbards
  
  At this point it's clear there are tons of more famous and less accessible books on this subject :) I won't list them because if you are at this point of math development you can definitely find them yourself :)
  
  From here you can graduate to studying category theory, differential geometry, algebraic geometry, more advanced texts on combinatorics, graph theory, number theory, complex analysis, probability, topology, algorithms, functional analysis etc
  
  Most listed books and more can be found on libgen if you can't afford to buy them. If you are stuck on homework, you'll find help on [MathStackexchange] (https://math.stackexchange.com/questions).
  
  Good luck.

Four pieces of advice:

1. To understand vague sounding theorems, try to reduce them to a case which is easy to draw. Most of the theorems are tough to grasp precisely because they are generalized to the greatest degree possible. Visualizing a more concrete case is a great way to 'feel' or 'understand' what a given theorem is really saying.
   
   
2. Get a supplemental book other than Rudin which has more exposition as a way to help pump your intuition. For a first course, I am a big fan of Ross:
   
   http://www.amazon.com/Elementary-Analysis-Calculus-Undergraduate-Mathematics/dp/1441928111/ref=sr_1_1?ie=UTF8&qid=1314638922&sr=8-1
   
   
3. Don't get too caught up in the logic. It is really easy to get so worked up about the minutiae of a proof that you start forgetting what, exactly, it is that you are trying to prove. Focus instead on getting a feel for why something should be true, and treat the proof as a way of arguing why it is true. When approaching it this way, I find that the details tend to work themselves out.
   
   
4. And if you get frustrated, just remember, most of the concepts you are trying to prove had been around and generally accepted for years before they had a rigorous foundation. You will literally be covering 2000+ years of math over the course of a semester. As long as you understand the larger picture and get a lot of practice doing proofs, the course will be accomplishing everything an introductory analysis course should.
   
   Analysis is a profound and beautiful subject. Enjoy it, and good luck!

If you're interested in Fourier series in general, I'd recommend a couple of different books. They all contain these results (some contain more constructive versions than others).


[Stein and Shakarchi's Fourier Analysis: An Introduction] (http://www.amazon.com/Fourier-Analysis-Introduction-Princeton-Lectures/dp/069111384X) is probably the most accessible book I can think of. It doesn't assume much analysis background, and it's a pretty easy read. It contains all the classical goodies you should see on Fourier analysis and Fourier series without having to use any measure theory. It also springboards into the 3rd volume in this series, which is on measure theory.



Sticking with the classical camp but adding in a bit of measure theory and functional analysis, there's Katznelson's An Introduction to Harmonic Analysis and the infamous Zygmund Trigonometric Series. Zygmund is an exceedingly comprehensive introduction to Fourier series at the beginning graduate level. And I do mean comprehensive. It was published in 1935, and it's a fair bet that it captured close to everything that was known about convergence results concerning Fourier series at that time.


The last way I'd go (and I wouldn't really look at it until you have some background in the above) is Javier Duoandikoetxea's Fourier Analysis. The book makes very free use of measure theory and functional analysis. It also assumes a pretty good working familiarity with the theory of distributions (which it introduces at rapid speed).

This is slightly off topic, but if anyone is like me, and finds the style of rudin a bit too terse and magical, I'd recommend Apostol as a nice book in the same spirit, but (IMO) better written:

https://www.amazon.com/Mathematical-Analysis-Second-Tom-Apostol/dp/0201002884

It also has great, classic style cover art, and hardcover copies are well bound. Mine's goin on 15 years and is in good shape.

You need some grounding in foundational topics like Propositional Logic, Proofs, Sets and Functions for higher math. If you've seen some of that in your Discrete Math class, you can jump straight into Abstract Algebra, Rigorous Linear Algebra (if you know some LA) and even Real Analysis. If thats not the case, the most expository and clearly written book on the above topics I have ever seen is Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.

Some user friendly books on Real Analysis:

1. Understanding Analysis by Steve Abbot
   
   
2. Yet Another Introduction to Analysis by Victor Bryant
   
   
3. Elementary Analysis: The Theory of Calculus by Kenneth Ross
   
   
4. Real Mathematical Analysis by Charles Pugh
   
   
5. A Primer of Real Functions by Ralph Boas
   
   
6. A Radical Approach to Real Analysis by David Bressoud
   
   
7. The Way of Analysis by Robert Strichartz
   
   
8. Foundations of Analysis by Edmund Landau
   
   
9. A Problem Book in Real Analysis by Asuman Aksoy and Mohamed Khamzi
   
   
10. Calculus by Spivak
    
    
11. Real Analysis: A Constructive Approach by Mark Bridger
    
    
12. Differential and Integral Calculus by Richard Courant, Edward McShane, Sam Sloan and Marvin Greenberg
    
    
13. You can find tons more if you search the internet. There are more superstars of advanced Calculus like Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra by Tom Apostol, Advanced Calculus by Shlomo Sternberg and Lynn Loomis... there are also more down to earth titles like Limits, Limits Everywhere:The Tools of Mathematical Analysis by david Appelbaum, Analysis: A Gateway to Understanding Mathematics by Sean Dineen...I just dont have time to list them all.
    
    Some user friendly books on Linear/Abstract Algebra:
    
    
14. A Book of Abstract Algebra by Charles Pinter
    
    
15. Matrix Analysis and Applied Linear Algebra Book and Solutions Manual by Carl Meyer
    
    
16. Groups and Their Graphs by Israel Grossman and Wilhelm Magnus
    
    
17. Linear Algebra Done Wrong by Sergei Treil-FREE
    
    
18. Elements of Algebra: Geometry, Numbers, Equations by John Stilwell
    
    Topology(even high school students can manage the first two titles):
    
    
19. Intuitive Topology by V.V. Prasolov
    
    
20. First Concepts of Topology by William G. Chinn, N. E. Steenrod and George H. Buehler
    
    
21. Topology Without Tears by Sydney Morris- FREE
    
    
22. Elementary Topology by O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev and and V. M. Kharlamov
    
    Some transitional books:
    
    
23. Tools of the Trade by Paul Sally
    
    
24. A Concise Introduction to Pure Mathematics by Martin Liebeck
    
    
25. How to Think Like a Mathematician: A Companion to Undergraduate Mathematics by Kevin Houston
    
    
26. Introductory Mathematics: Algebra and Analysis by Geoffrey Smith
    
    
27. Elements of Logic via Numbers and Sets by D.L Johnson
    
    Plus many more- just scour your local library and the internet.
    
    Good Luck, Dude/Dudette.

Square one...

You should have a solid base in math:

Introduction to Calculus and Analysis, Vol. 1 by Courant and John. Gotta have some basic knowledge of calculus.

Mathematical Methods in the Physical Sciences by Mary Boas. This is pretty high-level applied math, but it's the kind of stuff you deal with in serious physics/astrophysics.

You should have a solid base in physics:

They Feynman Lectures on Physics. Might be worth checking out. I think they're available free online.

You should have a solid base in astronomy/astrophysics:

The Physical Universe: An Introduction to Astronomy by Frank Shu. A bit outdated but a good textbook.

An Introduction to Modern Astrophysics by Carroll and Ostlie.

Astrophysics: A Very Short Introduction by James Binney. I haven't read this and there are no reviews, I think it was very recently published, but it looks promising.

It also might be worth checking out something like Coursera. They have free classes on math, physics, astrophysics, etc.

First, please make sure everyone understands they are capable of teaching the entire subject without a textbook. "What am I to teach?" is answered by the Common Core standards. I think it's best to free teachers from the tyranny of textbooks and the entire educational system from the tyranny of textbook publishers. If teachers never address this, it'll likely never change.

Here are a few I think are capable to being used but are not part of a larger series to adopt beyond one course:
Most any book by Serge Lang, books written by mathematicians and without a host of co-writers and editors are more interesting, cover the same topics, more in depth, less bells, whistles, fluff, and unneeded pictures and other distracting things, and most of all, tell a coherent story and argument:

Geometry and solutions

Basic Mathematics is a precalculus book, but might work with some supplementary work for other classes.

A First Course in Calculus

For advanced students, and possibly just a good teacher with all students, the Art of Problem Solving series are very good books:
Middle & high school:
and elementary linked from their main page. I have seen the latter myself.

Some more very good books that should be used more, by Gelfand:

The Method of Coordinates

Functions and Graphs

Algebra

Trigonometry

Lines and Curves: A Practical Geometry Handbook

What do you want to do, though? Is your goal to read math textbooks and later, maybe, math papers or is it for science/engineering? If it's the former, I'd simply ditch all that calc business and get started with "actual" math. There are about a million books designed to get you in the game. For one, try Book of Proof by Richard Hammack. It's free and designed to get your feet wet. Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand/Polimeni/Zhang is my favorite when it comes to books of this kind. You'll also pick up a lot of math from Discrete Math by Susanna Epp. These books assume no math background and will give you the coveted "math maturity".

There is also absolutely no shortage of subject books that will nurse you into maturity. For example, check out [The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Grinberg](https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Princeton/dp/0691172935/ref=sr_1_1?ie=UTF8&qid=1486754571&sr=8-1&keywords=real+analysis+lifesaver() and Book of Abstract Algebra by Pinter. There's also Linear Algebra by Singh. It's roughly at the level of more famous LADR by Axler, but doesn't require you have done time with lower level LA book first. The reason I recommend this book is because every theorem/lemma/proposition is illustrated with a concrete example. Sort of uncommon in a proof based math book. Its only drawback is its solution manual. Some of its proofs are sloppy, messy. But there's mathstackexchange for that. In short, every subject of math has dozens and dozens of intro books designed to be as gentle as possible. Heck, these days even grad level subjects are ungrad-ized: The Lebesgue Integral for Undergraduates by Johnson. I am sure there are such books even on subjects like differential geometry and algebraic geometry. Basically, you have choice. Good Luck!

I can't speak to Gouvea, but in my undergrad p-adic course we used Katok which wasn't to bad. As long as your analysis and topology are reasonably well practiced than it should make a good companion text.

If you're interested, my professor for that class (Keith Conrad) wrote up some great handouts on various topics in p-adic analysis which can be found here under the Algebraic Number Theory section. I'd really recommend them.

>My first goal is to understand the beauty that is calculus.

There are two "types" of Calculus. The one for engineers - the plug-and-chug type and the theory of Calculus called Real Analysis. If you want to see the actual beauty of the subject you might want to settle for the latter. It's rigorous and proof-based.

There are some great intros for RA:

Numbers and Functions: Steps to Analysis by Burn

A First Course in Mathematical Analysis by Brannan

Inside Calculus by Exner

Mathematical Analysis and Proof by Stirling

Yet Another Introduction to Analysis by Bryant

Mathematical Analysis: A Straightforward Approach by Binmore

Introduction to Calculus and Classical Analysis by Hijab

Analysis I by Tao

Real Analysis: A Constructive Approach by Bridger

Understanding Analysis by Abbot.

Seriously, there are just too many more of these great intros

But you need a good foundation. You need to learn the basics of math like logic, sets, relations, proofs etc.:

Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers

Discrete Mathematics with Applications by Epp

Mathematics: A Discrete Introduction by Scheinerman

Hi, a similar question was asked a couple days ago. I recommend reading GOD_Over_Djinn's excellent explanation here: http://www.reddit.com/r/math/comments/1h2i9v/playing_around_with_an_idea_related_to_prime/caqgyd5 or my own comment here: http://www.reddit.com/r/math/comments/1h2i9v/playing_around_with_an_idea_related_to_prime/caqgh42. The best way to learn about p-adic numbers is of course to read a book about them instead of just looking at wikipedia or reading what random people on the internet have to say. I cannot recommend enough Robert's "A Course in p-adic Analysis" if you have a basic knowledge of topology and analysis http://www.amazon.com/Course-p-adic-Analysis-Graduate-Mathematics/dp/0387986693. If you're more interested in p-adic zeta functions etc. look at Koblitz's "p-adic Numbers, p-adic Analysis and Zeta Functions" http://www.amazon.com/Numbers-Analysis-Zeta-Functions-Graduate-Mathematics/dp/1461270146/ref=sr_1_8?s=books&ie=UTF8&qid=1372366949&sr=1-8&keywords=p-adic+analysis. Although I haven't personally read it this book here also seems to be a more elementary introduction: http://www.amazon.com/p-adic-Numbers-Fernando-Quadros-Gouvea/dp/3540629114/ref=sr_1_3?s=books&ie=UTF8&qid=1372367005&sr=1-3&keywords=p-adic+analysis. The first 2 I know you can find pdfs of online. I don't know about the third. Alternatively, p-adic numbers are covered in a less technical sense in Bartel's notes on number theory here: http://homepages.warwick.ac.uk/~maslan/numthry.php. I haven't looked at them yet but I can say that his notes on representation theory are very good.

Any other construction I can think of aside from what I linked requires group theory or topology so its kinda hard unless you have a background in these subjects.

Edit: Having skimmed through Bartel's notes: they are an excellent introduction to p-adic numbers and he thoroughly covers them and their applications. I do recommend it.

>How was your experience with your first analysis course?

Unusual. My Calc 3 professor introduced me to another professor that was looking for undergrads to do research with, and over summer I did a reading course with him, for which he chose p-adic analysis (out of this text). This was definitely a "throw them into the deep end and see if they can swim" experience, and in retrospect I wrote a lot of proofs that didn't quite make sense, but ultimately I got a lot out of it. Then in fall, I took real analysis 1 and essentially laughed my way through it.

Some things that I think contributed to this going well: I had been out of school for a few years and supporting myself by tutoring math, so I had developed a bit of mathematical maturity in that I could usually tell when a solution was airtight or not. I also had very little else on my plate at the time, so I was able to work on the one subject for about 20 hours per week for 10 weeks straight, and had lots of opportunities to ask questions and discuss things with the professor.

To understand the FFT you first need to understand the DFT. The FFT is just a clever way of going about the DFT to reduce the runtime from O(n^2) to O(n log n).

Maybe try googling around for Cooley-Tukey explanations?

I did pick up a book a while back called "Who is Fourier?" (https://www.amazon.com/Who-Fourier-Mathematical-Adventure-2nd/dp/0964350432/ref=sr_1_fkmr0_2?ie=UTF8&qid=1480353431&sr=8-2-fkmr0&keywords=who+is+joseph+fourier) that did a surprisingly good job of explaining Fourier series and transforms. I am pretty sure there is a section in the book for the FFT, I can check when I get home if you're interested.

Many of the graduate level macro textbooks (e.g. SLP) are best read after getting some experience in mathematical analysis (at least one class at even a low level like that of Ross would probably make a huge difference in your comfort level with the material).

Romer's text is the first thing that comes to mind as something that is at a higher level than undergrad texts, but simple enough that someone well-versed in econ and differential equations can jump into without having a background in analysis.

If you're interested in autoreggresions, maybe a text on time series econometrics would be best.

I’m deeply in love with the scientist and engineers guide to dsp I’m doing audio dsp work and this was a big help, but it applies equally to shaders. It is not a how to guide but a core fundamentals book.

Along the same lines who is Fourier is a great but very quirky introduction to the fast Forier transform.

And Designing Audio Effect Plug-Ins in C++ is a good base intro to proper audio dsp development.

Historically, mathematicians had a goal of obtaining all integrals of rational integrands as rational expressions, rational expressions that would be given explicitly (or in closed form) in terms of elementary expressions. However, it was realized eventually that such a goal is a hopeless goal, one that is not possible in the traditional sense, and that the traditional artificial restrictions imposed on elementary analysis are thus unjustified.

They were brought to this realization most popularly by the elliptic integrals, integrals of rational expressions (rational expressions with the square root of a polynomial of the 4th or 3rd degree as an argument) which does not resolve itself into an explicit elementary expression by the methods of substitution or integration by parts.

Instead, due to greater rigor as gifted to us by the field of mathematical analysis, we were thus able to justify processes of approximations with a level of confidence and certainty that was not offered before.

As an elementary example, from Richard Courant's and Fritz John's "Introduction to Calculus and Analysis I", page 410 - 411, an integral expression for the time period of an ordinary pendulum was obtained, it is an elliptic integral, which means, we cannot proceed by way of a simple transformation of the independent variable ("method of substitution") or by breaking the integral apart into smaller parts by way of integration by parts and still hope to obtain a simple explicit elementary expression.

So, instead, there is an expression in the integrand, being, 1/√[1 - u^2 sin^2 (𝜃/2)]. For sufficiently small values of 𝜃, we find that the expression is arbitrarily close to the value 1, and therefore, this entire expression in the integrand was reduced to the factor one, allowing us to approximate the elliptic integral in sufficiently small intervals of 𝜃.

Noting that, the margin of error must be calculated (as was done so in the book). At least the physicists now have an expression for the time period of an ordinary pendulum - an imperfect approximation.

.

Admitting defeat:

Over the decades and eventually, centuries, mathematicians decided to allow functions as integral expressions without requiring always that they must be solved explicitly in terms of elementary expressions due to the convenience offered, in fact, the famous Gamma function is exactly the example, a function that is usually expressed as an integral (and, of course, don't forget about the elliptic integrals).

In the end, not all integrals are meant to be solved the same way that an integral of an elementary polynomial is, this philosophy is not merely isolated to that of integral calculus, as its analog can be found in differential calculus as differential equations.

Hartshorne's Geometry: Euclid and Beyond is a much more readable book compared to his other well-known work.

In addition to Needham, I've heard very good things about Remmert's Theory of Complex Functions for its use of history and Wegert's Visual Complex Functions for its visual approach to complex analysis, similar to but perhaps more rigorous than Needham. Kenji Ueno's three-volume A Mathematical Gift is similar in its intuitive explanations, but it covers various topics in mathematics as opposed to just complex analysis and can act as a nice introduction or as light reading (yes, he has another three-volume work on AG). I can also recommend Foundations and Fundamental Concepts of Mathematics by Howard Eves for its breezy overview of the foundations of mathematics, for anyone interested in that.

Edit: Links

There are also some nice books on calculus, such as Excursions in Calculus by Robert M. Young and New Horizons in Geometry by Mamikon A. Mnatsakanian and Tom M. Apostol (of Calculus and Analytic Number theory fame).

My favorite book on problem solving is Problem Solving Through Problems. There's an online copy, too. (I recommend you print it and get it bound at Kinkos if you intend to seriously work through it, though. This type of thing sucks on a screen.)

How To Solve It is another popular recommendation for that topic. Personally, I only read part of it. It's alright.

I can recommend other stuff if you tell me what level of math you're at, what you're interested in learning, etc.

I can't really speak for number theory, but Analysis can be quite difficult to understand at first, especially with no one to explain it to you. Eventually it starts to click, and becomes a lot easier in some ways, but it can take a decent bit of perseverence. You definitely can teach yourself them, but you can't be afraid to ask for help (either on /r/learnmath, or over on math stackexchange). If you don't ask when you don't understand it, it is really difficult to progress on your own sometimes.

As for books, we used Haggarty's Fundamentals of Mathematical Analysis when I first learnt it, and while it is very thorough, and intro level, it's also relatively pricey for what it is (which is basically only the very basics - first year analysis for math students in the UK). It is definitely a good book for beginners though.

I would recommend the following two books:

1. "How to Prove It" by Daniel Velleman.
   
   
2. "Understanding Analysis" by Stephen Abbott.
   
   The first book introduces most of the topics in the book that you linked, and was what was used in my Foundations of Mathematics class (essentially the same thing as your class).
   
   Understanding Analysis, on the other hand, is probably the perfect book to follow up with, since it is such a well-motivated, yet rigorous book on the analysis of one real variable, that you may, in fact, become too accustomed to such lucid and entertaining prose for your own good.

this book introduces both the open set and epsilon delta formulation at the beginning, allowing you to learn from both perspectives. In addition to that, it's simply a wonderful textbook, the best i've seen for a first course in Analysis to non honors students.

> This seems very confusing to me, as it is defining p-adic expansion of numbers in terms of p-adic numbers ...

This is just a hand-wavy, intuitive explanation of what p-adic numbers look like. The fact is that once you formalize everything about the [p-adic valuation](http://en.wikipedia.org/wiki/P-adic_valuation) and the p-adic numbers, it turns out that every p-adic number has the series expansion that you mentioned.

> For instance, why, in the p-adic world, are positive powers of p small, and negative powers large? It seems like a prime number to a large power would be large, no?

When dealing with p-adic numbers, you have to forget all your intuition about the usual notions of absolute values and ordering of the real numbers, since they don't apply. Everything in the p-adic world is based on the p-adic valuations, which give their own topologies and notions of size. The p-adic topologies are very different from the topology on R. For example, any point within an open ball in the p-adic numbers can be considered that ball's center. Quirky things like this make it initially hard to grasp the concepts of p-adic numbers and their associated arithmetic, but once you practice working with them enough, they start to make sense.

> How does the limit of the sequence that they're talking about equal 1/3?

This again has to do with the fact that convergence in the p-adic topology is different from convergence in the usual Euclidean topology.

Some good resources for learning more about p-adic numbers are the following:

1. Gouvêa, Fernando Quadros, p-adic Numbers: An Introduction (Amazon, SpringerLink)
   
2. Koblitz, Neal, p-adic Numbers, p-adic Analysis, and Zeta-Functions (Amazon, SpringerLink)
   
3. Robert, Alain M, A Course in p-adic Analysis (Amazon, SpringerLink)
   
4. Serre, Jean-Pierre, A Course in Arithmetic (Amazon, SpringerLink)
   
   For me personally, learning general valuation theory was very useful for understanding p-adic numbers.

I liked Abbott's book but the one that I found genuinely amazing was Haggarty

It's very hard to find in print, but there are definitely some illicit pdfs and djvus out there if you know where to look

There are very few true textbooks - i.e. books designed to teach the material to those who don't already know the classical versions - written in this style.

• Bridger: Real Analysis - A Constructive Apporach
  
• Bell: A Primer of Infinitesimal Analysis - not constructive per se, but an exposition of non-classical mathematics.
  
• Vickers: Topology via Logic
  
• Lombardi: Commutative Algebra - the odd-one-out in that one does need a good working knowledge of classical commutative algebra to read the book, but it presents a whole lot of genuinely new material textbook-style.
  
• Taylor: Practical Foundations of Mathematics - Chapter 3 is an intro to constructive order theory.
  
• Troelstra: Choice Sequences - again, hardly constructive, but a good exposition of an important part of intuitionistic mathematics.
  
  While we're at it, a quick skim through the algebra chapter of Troelstra: Constructivism in Mathematics, vol. 2 should explain why there are no textbooks on abstract algebra written in the purely constructive tradition.

I hear that Rudin's book is pretty dense, so initially, I won't be using it, though I'm not entirely familiar with Spivak/Rudin beyond the comments on Amazon/Reddit.

Instead, I'm reading from Ross and [Bartle] (https://www.amazon.ca/Introduction-Real-Analysis-Robert-Bartle/dp/0471433314) right now, which I hear are good books for people starting out in Analysis. As I progress through the series, I might start teaching from Rudin and a variety of other sources.

First you have to decide what kind. The easiest to understand IMHO is Nelson's IST, and the best introduction to that is the first chapter of an unpublished book by Nelson.

Robinson-style NSA doesn't have logical/set theoretical foundations. It has model theoretic foundations. And there isn't just one Robinson-style NSA. Rather there are lots of different ones based on different models.

If you want to know more about different axiomatic NSA the book for that is Nonstandard Analysis, Axiomatically.

Although I have a bunch of books on Robinson-style NSA, I don't recommend any of them (for me the subject is very confusing compared to IST). Goldblatt is OK.

In the grand scheme of math: jack shit. But who's to stop you after 2 months of studying?

What do you know so far? Are you comfortable with inequalities and math induction?

Check out the books below for a nice intro to Real Analysis:

How to Think About Analysis by Lara Alcock.

A First Course in Mathematical Analysis by D. A. Brannan.

Numbers and Functions: Steps to Analysis by R. P. Burn.

Inside Calculus by George R. Exner .

Discrete And Continuous Calculus: The Essentials by R. Scott McIntire.

Good Look.

If you are interested Complex Analysis, Needham's book is a great read!

p-adic Analysis Compared with Real is the main text I've used. If you haven't taken real analysis, then parts of the book may be lost on you, but there are definitely large sections that will still be accessible.

It has lots of exercises as well as "Answers, Hints, and Solutions for Selected Exercises" at the end.

By the way, is that the Hughes-Hallett et al book? If so, it's notoriously mediocre.

If you want another resource, these are pretty good and can be bought used fairly cheap:

• Lang: https://amazon.com/dp/0387962018/
  
• Ash and Ash: https://amazon.com/dp/0879421835
  
  Or you can just head to YouTube. Whatever works for you.

Tbh, I can't tell you from personal experience because after I got into university I stopped preparing for competitions and use my free time to study some other math instead.

Anyway, here are some books that I've heard are good for Putnam-esque preparation.

Putnam And Beyond

Problem-Solving Through Problems

Problems From The Book

Straight From The Book

If you want something a bit more in depth than Wikipedia, baby Rudin is the standard analysis textbook and can be had for about ten dollars on Amazon. It's pretty dense though.

https://www.amazon.com/Principles-Mathematical-Analysis-Walter-Rudin/dp/1259064786/ref=pd_sbs_14_t_0?_encoding=UTF8&psc=1&refRID=2JHRTDF15JZR0HJF56GH

If you're serious about learning about Fourier Analysis, the book Who is Fourier?: A Mathematical Adventure is an excellent and very accessible introductory text on the subject.

/u/3blue1brown Have you seen Wegert’s book Visual Complex Functions (site, gallery, amzn)?

He has a number of nice phase portraits of the Zeta function and of its critical strip, etc.

You can get a cheap international 3rd edition very easily. That would be better than looking for an errata for the previous editions.

True, I do seem to be asking for a very specific plan. Is it this book?

https://www.amazon.com/Who-Fourier-Mathematical-Adventure-2nd/dp/0964350432

I've always liked Ross' Elementary Analysis

George Bergman's companion exercises to Rudin's textbook for Chapters 1-7.

Roger Cooke's solutions manual for Rudin's analysis

A subreddit devoted to Baby Rudin with further resources in the sidebar.

Tom Apostol's textbook

I find that Rudin is to Analysis textbooks what C++ is to programming languages. A little difficult at first, but with so many auxiliary sources that it becomes one of the best texts to learn from in spite of this.

> In particular, I am struggling with concepts of topology and mappings between metric spaces. I just cannot visualize what is going on in an intuitive way.


I think Apostol's text should work for you. The first few chapters use a lot of point-set topology language with a fair number of examples & proofs...that should hopefully address the issues you mentioned.

Who Is Fourier? A Mathematical Adventure

https://www.amazon.com/Who-Fourier-Mathematical-Adventure-2nd/dp/0964350432

You are missing Abstract Algebra that usually comes before or after Real Analysis. As for that 4chan post, Rudin's book will hand anyone their ass if they havent seen proofs and dont have a proper foundation (Logic/Proofs/Sets/Functions). Transition to Higher Math courses usually cover such matters. Covering Rudin in 4 months is a stretch. It has to be the toughest intro to Real Analysis. There are tons of easier going alternatives:

Real Mathematical Analysis by Charles Pugh

Understanding Analysis by Stephen Abbot

A Primer of Real Functions by Ralph Boas

Yet Another Introduction to Analysis

Elementary Analysis: The Theory of Calculus

Real Analysis: A Constructive Approach

Introduction to Topology and Modern Analysis by George F. Simmons

...and tons more.

You may be thinking of Courant's original Differential and Integral Calculus in two volumes. What I have is the solutions to problems from the updated and expanded to about twice the size, Introduction to Calculus and Analysis, in three volumes.

introduction to calculus and analysis (3 book set) by courant and john:

volume 1

volume 2 book 1

volume 2 book 2

Goldblatt can.

Here is the mobile version of your link

Both his books are considered to be very tough (or at least, the problems are). OTOH, I've often heard that if you can somehow get through them, then you'll know analysis really well.

If the first Rudin is too much for you (or rather, too terse and you're wondering what the whole point is), try Kenneth Ross. It's quite easy to follow, and the problems aren't too easy.

The 1st Course In Math Analysis by Brannan

Analysis I by Terrence Tao

Yet Another Intro To Real Analysis by Bryant

Understanding Analysis Stephen Abbott

Elementary Analysis: The Theory of Calculus Kenneth A. Ross


Metric Spaces by Robert Reisel

A Problem Text in Advanced Calculus by John Erdman. PDF

Advanced Calculus by Shlomo Sternberg and Lynn Loomis.PDF

I'm asking someone to show me how to do the problem. So I know how to do it. I learned a lot from

ss0317 2 points 3 hours ago
You can solve this by inspection. The fundamental frequency is the lowest first order harmonic of all of your sin and cos terms.
100pi = 2pif0 -> f0 = 50Hz, or w0 = 100pi rads/s.
Period = 1/f0 = 20 ms.
Might I recommend this book: http://www.amazon.com/Who-Fourier-Mathematical-Adventure-2nd/dp/0964350432/ref=sr_1_fkmr0_3?ie=UTF8&qid=1465228053&sr=8-3-fkmr0&keywords=who+is+joseph+fourier