The Best Math Textbook for Everyone

I discuss one of my favorite math textbooks, Fourier Analysis: An Introduction by Elias Stein and Rami Shakarchi. I recommend this book to everyone in a STEM field who knows multivariable calculus and has some familiarity with ODE and linear algebra.
This book is excellent for initiating students into analysis. Students need not wait until after a semester of real analysis to learn from this book.
Amazon link: amzn.to/3yfnzta
(I get a small commission for purchases made with this link.)
Check out my other math book recommendation videos!
The 3 Best Books on Complex Analysis • The 3 Best Books on Co...
Elliptic Curves: Good Books to Get Started • Elliptic Curves: Good ...
Check out my series Tricky Parts of Calculus, a series that covers the subtle and difficult topics in calculus that are usually missed in a calculus class. See the whole playlist here: • Tricky Parts of Calculus
And check out my podcast for great conversations about math and other topics: • Daniel Rubin Show, Ful...
Let me know what you think of this book in the comments, or if you have a different preference for treatment of Fourier analysis.

[@robertforster8984]

I used this book along with baby Rudin and My teacher’s (Patrick Gallagher) hand written notes in my analysis class. My teacher actually took courses with Lars Ahlfors in the early ‘50s using Ahlfors own Complex Analysis book. His Phd advisor grandfather was G.H. Hardy. I was lucky to have known and done my senior thesis under him.

[@DanielRubin1]

Pat Gallagher was actually on my PhD thesis defense committee.

[@chefrich1279]

Stein and Shakarachi's Book on Measure Theory; particularly chapter 1 is *chef's kiss*. No abstract sigma-algebras, just a intuitive explanation of the Lesbesgue measure and a development of integration theory. Then and only then do you tackle more general measures.

[@DanielRubin1]

I agree. As an undergrad I took a course that followed Stein and Shakarchi's Real Analysis, and I really enjoyed that book as well.

[@indianatlarge]

this is sheer nostalgia.. i am going to buy this book again and read all over again,

[@leodu561]

As a current Johns Hopkins student who also went through dozens of pure math books by now (and also suffered in grades in high school because I was way too into pure math), I am very very moved by your story. I bought the Princeton four part lecture series for a bargain before I came to USA. I didn't finish those but for the parts I did read, they were amazing. These are some of my prized books in my now quite large math books collection.

[@DanielRubin1]

Nice to hear that. Curious to know what you think of my Math Major Guide video. My views about math have changed quite a bit since undergrad.

[@ronykroy]

could you please elaborate on the "Princeton four part lecture series" What is that..?

[@leodu561]

​@@DanielRubin1 I watched through that video and I have a looot of thoughts so I'll write something in that video :)

[@leodu561]

@@ronykroy This is the first book in a series of four books which form the Princeton Lectures in Analysis: Fourier Analysis, Complex Analysis, Real Analysis (with measure), Functional Analysis. Buying the entire collection can be incredibly cheap if you're in certain parts of the world.

[@anthonyymm511]

Took a course based on this book a semester before I took measure theory and functional analysis. Glad I did it that way and not the other way around.

[@akashsudhanshu5420]

Can you explain me what you felt different? I have Measure and Integration next to next sem. And Fourier Analysis here has Functional Analysis as per requisite.

[@nini_kys]

17 is insane, you must be very talented. Thanks for the great vid. I'm in my second sem of analysis as a third year undergrad studying CS and math. This vid is really insightful to me as I get to pick a textbook to read with graduate students and I wanted to pick either this one or Grafakos as they recommend that for students who've already have taken analysis.

[@felixchengtk]

I have to check this book out from the library! My favorite Fourier Analysis book is G Folland's Fourier Analysis and It's Applications.

[@atharvmishraiiit-k]

Wow..... you did this this the age of 17 its quiet impressive !!!

[@julianwilson9919]

Goes well with Bressoud's A Radical Approach to Real Analysis. Bressoud starts with Fourier to motivate the development of real analysis.

[@ulysses_grant]

Great one! I'll definitely check this one out! As far as I had been using it, Susanna Epp - Discrete Mathematics with Applications is my current best of all time math book.

[@chrislang7432]

An excellent text, which I've used as an instructor; beautifully put together and crystal clear throughout. There are many books on Fourier analysis, but this text really does stand out.

[@skwalka6372]

How well does this book cover the Gibbs phenomenon? This is a pesky issue when it comes to practical applications. Thanks!

[@monkeeseemonkeedoo3745]

Wasn't expecting to see this in my recommended. Thanks for sharing, will check it out

[@DanielRubin1]

Hope you enjoy it!

[@karthicka1324]

Great one! I feel like a book walkthrough, I wish we get to know more of your experiences and notes sharing in future

[@jmguevarajordan]

Congratulations for your review. That book is indeed the best one in the Princeton's serie on analysis by Stein and Shakarchi. However, Stein's presentations of math analysis is not the easiest or the most pedagogical one. Of course, once you know the material presented in Stein's books, those books become excellent references.

[@DanielRubin1]

The Stein and Shakarchi books are certainly not easy, but I think they are written to be very clear and useful to students, not just to experts. And the exercises are really excellent for training undergraduates to become analysts.

[@realcirno1750]

I think you should change the title. This textbook is for upper undergraduates, real analysis theorems (Weierstrass M test, uniform limit theorem, Heine-Cantor/Borel theorems), are invoked without dwelling on their proofs, this alone limits the audience of the book to the 1% of the population who decide to major in math or maybe physics, nevermind the fact that the content itself is pretty difficult

[@nini_kys]

Yeah I assume that the title of this video is meant to say... "for Everyone normally introduced to Lebesgue integration in their courses"

[@turdferguson3400]

My favorite math book of all time is Probability Theory: the Logic of Science by ET Jaynes.

[@MKDSLeone]

omg same here

[@thcoura]

If I may share a word of wisdom. You traveled a long path in education to reach your journey in graduation. So, if you read a book and don't get something you are not dumb or stupid. The book is the problem. Look for others about the topic. There are many ways to explain a subject and no author is perfect.

[@monkeeseemonkeedoo3745]

@@ElderEric I had a similar reaction. Among many things, math taught me I should be careful about making general statements

[@zack_120]

Wrong ! It's the teachers' problem instead.

[@chaeyoungswifey4153]

@@ElderEric I mean it's not fully the persons fault either, some ppl sure, but others genuinely try and don't do well🤷‍♀️

[@nickharrison3748]

@@zack_120 ..yes..our understanding depends on teacher's imagination

[@JohnSmendrovac]

I totally disagree

[@mathmusicstructure]

I recommend A. P. French's "Vibrations and Waves" from the MIT series as an introduction to the physics side. The book from the video and French would be a very effective combination.

[@billsmith7673]

Thanks for the recommendation. Just found book.

[@jayantshaq]

Great video. Your love for the book and the subject really shines through.

[@bm-br3go]

For anyone that really likes the contents of this book and is interested in some analysis closely tied to engineering applications, I HIGHLY recommend the book "Foundations of Time-Frequency Analysis" by Karlheinz Gröchenig. It's a phenomenal introduction to short-time Fourier transforms, gabor frames, modulation spaces, and it has a little bit of wavelet analysis as well.

[@DanielRubin1]

That book sounds really interesting. I'll check it out.

[@DavidMFChapman]

Fourier analysis is fundamental to the study of most physical processes. Essential stuff!

[@parkjeremie]

Great video! I'm currently taking a course in multivariable calculus and am looking forward to learning this subject afterwards.

[@dijonstreak]

loved this expo.....congrats on a very cool expose'....loved your sincere enthusiasm...great job buddy...the book s expensive online......!!

[@DanielRubin1]

Thanks so much!

[@antoniomantovani3147]

This on Fourier it's the reference for every languages for math and physics

[@portreemathstutor]

I think I will buy it. I was halfway through reading an online Fourier Analysis book when it was removed from the library so I lost access to it.

[@michealjackson1334]

Butzer and Nessel Fourier Analysis and Approximation

[@angadvishwakarma35]

Can you suggest any book for proof for set theory and function.just like to prove any mapping injective without any algebraic function.

[@valor36az]

Great review, will look to get the series

[@DanielRubin1]

Hope you enjoy it!

[@johngough2958]

I used to like Walter Rudin's books - very similar to Stein and Shakarchi's series which I'll need to look into. Reed and Simon's book are my favourite, but it heavily depends on what you are first exposed to and what your interests are.

[@alfredthepatientxcvi]

Thank you for proposing us this amazing book :)

[@jessewolf6806]

I don’t like the fact that the proofs do not end with QED but are embedded within the general prose; that makes it difficult to scan the book quickly.

[@mimmyjau]

great content. please make more of these!

[@DanielRubin1]

Thanks! More to come!

[@macmos1]

What does it mean by " where tau > 0 is a constant equal to the coefficient of tension of the string" Are the units of tau not unit-less? yn-yn+1 / h is a unit-less factor, but this is implying that tau has units of Newtons.

[@DanielRubin1]

Tau has units of force. It is the magnitude of the force that the string exerts on itself at any small segment that keeps the string together.

[@mhermarckarakouzian8899]

Interesting. Would you say the same for their Functional Analysis book? ‘Cause I need a good book for that at the moment. And your “trains you to be an analyst” selling point I could definitely use. I just need functional analysis more urgently at the moment, but I’ll check out the Fourier series book sometime down the line for sure.

[@NimrodTargaryen]

Congratulations, like your approach of maths to real world problems

[@surajmandal_567]

The beauty of mathematics is unmatched. ❤

[@jamesedward9306]

Thumbnail: "The best math book - NO QUALIFICATIONS NECESSARY" Rubin's comment: "I recommend this book to everyone in a STEM field who knows multivariable calculus and has some familiarity with ODE and linear algebra." Right.

[@neroresurrected]

Thank you! Glad someone pointed that out on this guy’s video here. Smh.

[@renscience]

Books like these are for those at a college level involved with engineering and sciences. A good precursor to this would be “Who is Fourier?” By the Transnational College of LEX. They also have a great book on QM. They teach children this stuff using these books!

[@Edward-bm7vw]

I'm shit at math but would really like to get better. This sounds like it may be a bit advanced for me

[@marke.8321]

“The best math book, no qualifications necessary”…. “Everyone should do these problems in the first chapter, I think I covered them in part 4 of my series on the tricky parts of calculus” Ok fine you must have meant no qualifications on “best math book”.

[@zack_120]

Nature doesn't pretend or hype. It does everything in the direct, simplest way. So all knowledges of nature including math, the nature's language, are simple and easily understandable. If not, it's the book writer or/and teacher who didn't do their job well, especially those who only tell the student What but not Why/How, the knowledge required to 'understand' but not only to 'know' which any fool is capable of. The reason being so is that either they themselves don't 'understand' or fail to explain to you.

[@jacobvandijk6525]

WHAT IS BEST IS A PERSONAL MATTER.

[@capefry8323]

Would this book work for an EE student who has taken differential equations plus linear algebra, multivariate calc, but currently no higher math beyond that?

[@harrydiv321]

Hey I know this is late but probably not, I recommend an introductory real analysis book first e.g. Abbott

[@James-bv4nu]

Fourier series, as well as Taylor series, when expanded on a continuous, physical function, should converge to a point on that function, always, and for any point on that function. Otherwise, what good are the series? Correct? (continuous meaning no jumps) (physical function means area under curve is finite) (for physical function with jumps should converge to midpoint between the jump points)

[@DanielRubin1]

I disagree with this philosophy. It is one thing to speak of the intuition or aspiration or purpose of the introduction of a certain model or something like the representation of a function by a series. But having set down a precise mathematical meaning to our model, we can then use purely mathematical methods to determine the conditions under which our model achieves its purpose. Detailed convergence theorems and counterexamples are not just mathematical curiosities. These types of results tell us about the consequences and applicability of models like PDE to physical phenomena, for example with regard for propagation of singularities. See the discussion about how mathematicians and physicists view PDE differently in my podcast episode with Phil Sosoe. And representations can convey plenty of important information even if they do not converge in some particular sense; take for example asymptotic formulae.

[@justsomeone4797]

Interesting video and thanks for the recommendation! I have a question, what book would you recommend to an aspiring math student who wants to sharpen his mathematical thinking and problem solving ability?

[@DanielRubin1]

I really like Proofs from THE BOOK, by Aigner and Ziegler. It introduces a lot of important math that a beginner can pick up, and all the proofs are excellent for seeing what's going on and learning the important techniques. For trying out solving problems on one's own, I remember enjoying Paul Zeitz's The Art and Craft of Problem Solving.

[@californiadreamin8423]

@@DanielRubin1 Hello and thanks for the video and the excitement it is giving me. I graduated 50 years ago in Aeronautical Engineering and Design , and this analysis was essential in Stability and Control which was one of my final year subjects, and the one I was most in tune with…..and I never ever used it again !! After 10 years as structural mechanical designer, I retrained as an airline pilot but my time at university gave me a foundation of confidence in my flying career. Now retired , I’ve been working up my maths skill again to keep me mentally active, filling in gaps in my knowledge which 50 years ago always left me uncomfortable, but Fourier Analysis is one of my goals. I’m from across the pond. Edit: You mentioned a situation which had blown up in your face, and I had a similar situation when we were all required to give a presentation before our course and lecturers regarding our final year projects. I was the only student to receive a “rubbish” when I mention that the Southwell Plot worked for the sandwich material I was testing. I was so humiliated that I went away determined to vindicate myself. I couldn’t find anything after much searching in the library, so I was forced to go back to first principles and prove it….and I did, so it gave me, and still does, great pleasure in my throw away line in my thesis….see Appendix A for proof 😊. I forgot to mention how I really enjoyed Laplace Transforms.

[@davidwright5719]

Do you have a recommendation for a book on Fourier Analysis on manifolds besides R^n? For example, on the sphere?

[@carlobenedetti2407]

This is an outstanding book

[@anthony9656]

Now I may have to pick this book up. Would you recommend reading the introduction to Fourier series by Maxime Bocher?

[@SmithnWesson]

How do you feel about the print quality of your copy of the book? I just ordered a copy of the book. I think the paper is too yellow, and the print is not dark enough. Maybe I just got a bad copy. Or maybe they are all like this?

[@CrazyShores]

Hi, is it readable if i have real analysis, multivariable, and vector analysis, abstract linear algebra and a bit of abstract algebra?

[@f.osborn1579]

39 seconds...I’m sold

[@papsaebus8606]

what are the prerequisites for this? if you're completely self taught, would you recommend to go through this after doing courses on linear algebra and calc 1-3?

[@DanielRubin1]

Yes, Linear Algebra and Calculus 1-3 are the key prerequisites for this book. It would help to have a little experience with Ordinary Differential Equations, but not a lot is needed, maybe just the theory of linear equations with constant coefficients. I think that the relevant concepts (like uniform convergence) from a course in Real Analysis can be picked up from this book without needing that course first, and it's better to learn those concepts here anyway since they appear in a concrete context. The only other prerequisite is a willingness to engage with some difficult problems and give good proofs.

[@RobertoSuarez2444]

Then this textbook is NOT for everyone.

[@---tq2bo]

Do you recommend this particular book in the series or the entire series as well? are they meant to read in order? I'm asking cause I'm currently going through Rudin but I just don't feel like I'm learning anything

[@DanielRubin1]

As an undergraduate I went through their Complex Analysis and Real Analysis books, but I haven't spent much time with part 4 on Functional Analysis. All the books are well-written and have great exercises. They can be read in order and do tie together, but it's not necessary. There are a couple other complex analysis books I recommend based on choice of topics. Their Real Analysis text is about measure, Lebesgue integration, and Hilbert spaces; that's equivalent to a second semester of real analysis (after Rudin), and I think there's no rush for undergraduates to cover that material. See my comments in my Math Major Guide video. I think S&S's Fourier Analysis could help you understand what is the point of what's going on in Rudin and develop your techniques, but it's not actually a substitute for a first semester of real analysis. Check out Bressoud's A Radical Approach to Real Analysis for a historical approach that should also clear things up.

[@---tq2bo]

@@DanielRubin1 I will check that video out, thanks for the response.

[@gauravbansal2438]

Any good book for set theory , qhuch contains every set theory topic in details and with lot of examples , or books with different aspects

[@DanielRubin1]

I only know the minimal amount of set theory to work over the real numbers, so I can't recommend a book on this topic. But check out my podcast episode with set theorist/logician Joel David Hamkins. His books might be a good place to start.

[@gauravbansal2438]

Thnx ♥️

[@victorpopa8724]

Such a shame it's too expensive for me at this moment in time. I really want to dig in!

[@lowerbound4803]

The best review ever!!! Thank you😻

[@dannydai1]

Thanks, professor. Complex analysis and Fourier analysis, how can I learn it as an engineering guy?

[@gilloon9964]

Take a look at: Kreyszig Advanced Engineering Mathematics Zill Advanced Engineering Mathematics

[@BlackCat039]

Currently many books are supported by educational videos on the internet. It is much better to observe the concepts and their applications through animations. It is not difficult to enter the technological age.

[@joegarcia9082]

Do you think this would help if I'm taking a class in systems and signals for electrical engineering?

[@DanielRubin1]

Absolutely. I've never taken a course in that subject, but I know Fourier analysis to be the main theoretical tool, along with approximation theory and complex analysis. There's probably a lot of overlap between what's in Stein and Shakarchi and what you'll see in a text for your class, but this book has very careful statements of theorems and proofs, so you'll see how hypotheses matter and what counterexamples can come up, and it trains you in the techniques of analysis. Not everyone outside of math thinks these things are so important, but I think that without them the theory is too much of a black box.

[@kathieharine5982]

I have that book and it's three friends.

[@imranq9241]

What kind of interesting, worthwhile problems could i solve if i worked through the exercises in this book ?

[@DanielRubin1]

There are so many! There are problems that take you through the solution of the wave, heat, and Laplace equations on simple domains, plenty of convergence theorems for Fourier series and explicit constructions of counterexamples, there's the Heisenberg uncertainty principle, the reconstruction formula for the Radon transform, the theory of the gamma function. Then there's plenty of number theory: the values of the zeta function at even integers, various summation techniques, some problems about equidistributed sequences, character theory of finite abelian groups. The book is a gateway to work in all further fields of analysis and number theory, and also applications like signal processing.

[@Hatifnote]

All Books are introduction !!! All my life i read introductions.. That is to stay out the door without entering... But I think it's lucky to have this Big Book on Fourier Analysis.. I want go away over Fourier..

[@stumbling]

Based on the cover and font I'm guessing this was published around 2002.

[@billharm6006]

You sold a book. A good text for self-study can be hard to find, and your enthusiasm seems genuine. (And Amazon thanks you too). There is math for the sake of math. There is math for real world application. Occasionally, a book will bridge the two worlds. I hope this is one of those. Have occasionally "brushed up against" Fourier Analysis in the applied form. First with rotating equipment material condition monitoring (do we need to replace that bearing?) Then with passive sonar signal analysis (think, "waterfall display") and broad spectrum electromagnetic signal detection and analysis. Also with diesel engine vibration dampening (damper design) and internal combustion engine cam profile design (primarily for profile smoothing). Often the "Fourier part" is done with "pre-canned" FFT software (a good oscilloscope will have the option of time domain or frequency domain). Occasionally done with a "cookbook" FFT algorithm imbedded into ad-hoc custom code. Heck, even Excel has some support. Fourier has repeatedly been useful, but I've essentially been an "appliance user." I figure it's time to look behind the curtain (or under the hood, or whatever analogy you prefer).

[@DanielRubin1]

For me it went the other way. I didn't have much exposure to applied math as a student, but I find it really interesting now. Sounds like you've done some cool work. Enjoy the book!

[@omh5970]

Can this be used by a physics student?

[@DanielRubin1]

Absolutely! This book covers all the basic theory of Fourier series and the Fourier transform that a physics student needs, and covers their application to the wave, heat, and Laplace equations. The book is intended for an audience oriented towards mathematics as the text gives careful statements and proofs of theorems and the exercises are theoretical, but this shouldn't be an obstacle to physics students. But you will have to find another text if you're interested in signal processing or numerical methods of Fourier analysis.

[@pmcate2]

Sheesh you went through this when you were 17? I didn't even know what proof-based math was until I was 24.

[@DanielRubin1]

I had my mathematical awakening at the PROMYS program at Boston University when I was 16. There I got into the habit of writing clear proofs in the context of number theory. But there's something odd to me about the distinction people make about proof-based math and non-proof-based math. I've always thought that anything you assert you have to prove. I think Stein and Shakarchi's Fourier Analysis is great for students who just understand multivariable calculus and maybe some physics in a concrete way. It's a concrete subject that displays a lot of subtlety, so it's clear you have things to prove.

[@rogerbee697]

Hmmmm... perhaps I’ll brush up on add/subtract/multiply/divide first. 🙄

[@rohitnakkina2393]

where can i get solutions of problems in this book?

[@DanielRubin1]

Looks like there are some good resources on the web that cover many of the problems in the book. Check out studylib.net/doc/18485964/solutions-to-some-exercises-and-problems

[@michaeldelisieux]

Math is mysticism!

[@manleom2260]

NINJA!!..THIS IS THE BEST BOOK FOR YOUU!!..AN NO!..THE BEST EQUATION TO LEARN IS THE SCHRONDINGER'S EQUATION....

[@hassanhameed720]

Sir plz solve calculus 10th edition

[@rickdoesmath3945]

I want to become an analyst. Should i study fourier analysis?

[@DanielRubin1]

Yes, 100%, absolutely you should. In truth, you could have ended your sentence "I want to become a ______" with any specialty whatsoever within STEM and the answer would have been the same.

[@maalikserebryakov]

@@DanielRubin1 well that settles it then. *Goes to Library genesis*

[@meteor8076]

But what can you say about two other volumes ? Is it worth learning from them ?

[@DanielRubin1]

There are 3 other volumes in Stein and Shakarchi's Princeton Lectures in Analysis series: Complex Analysis, Real Analysis, and Functional Analysis. I have never used the volume on Functional Analysis, but the Complex Analysis and Real Analysis volumes are great for students just like the first volume. I discuss the Complex Analysis volume in my video on the best books for Complex Analysis. The Real Analysis volume, which covers measure theory, the Lebesgue integral, Hilbert space theory, and some ergodic theory and theory of Hausdorff measure and fractals, I hope to discuss in a future video.

[@meteor8076]

@@DanielRubin1 Wooow, very very interesting, thank you! I want to use Real Analysis by Stein and Shakarchi as a second course after Abbott

[@allanwrobel6607]

Enthusiasm is infectious.

[@hansbleuer3346]

Informativer Buchtipp!

[@Mrcometo]

I think you can use a protective spray to protect your beautiful notebook.

[@davidd355]

This book is amazing

[@pallan44592]

Just amazing.

[@justgivemethetruth]

How could this be a book for everyone .... no matter what their level in mathematics? How could that be? I've always been interested in Fourier Analysis, but I never got to where I felt I had an intuitive understand of what it is or how it works ... same with LaPlace Transforms. But, pretty expensive book too.

[@vajrapaniom7410]

Library. lol 🙂

[@manfredcaranci6234]

Oh, you don't know how correct you are! This is clickbait in its claim to be THE BEST MATH TEXTBOOK FOR EVERYONE. Absolutely disagree. I learned both Fourier series and transforms, as well as Laplace transforms, appropriate for an ENGINEER, from a book called "Signals, Systems, Communications" by B.P. Lathi (1965). I was able to apply what I learned about Fourier from this book in my 38-year career as an electrical engineer in the telecommunications industry. For the discrete Fourier series and transform, there's decent coverage in Oppenheim and Schafer's "Digital Signal Processing". Had I been forced to use the much-ballyhooed book described herein, I would've easily failed to grasp the first chapter. This book is for pure mathematicians, not practicing scientists or engineers.

[@Viroulu]

good book but it's too expensive. I will look for a used one

[@comradeedwin1006]

Is this book only for math majors?

[@nini_kys]

I mean no, but you need to analytic foundations of calculus demonstrated in Baby Rudin type Analysis class to get full understanding as theorems used here are proved there

[@gnuemacs1166]

Thank you

[@VCT3333]

I prefer Fiveier Analysis.

[@kitsurubami]

what's a good book for idiots like me? I have "calculus for dummies", but I struggle with it.

[@taniatania1842]

Hi can you answer me?

[@onlinemathslearning2439]

Online maths learning

[@malawigw]

its so expensive.

[@neroresurrected]

That’s the first hint that this book is not designed for the average lay person with very basic math skills in mind.

[@donbyzewski7377]

hay quá chừng luôn

[@rashie]

👍👍

[@kenichimori8533]

Proof

[@7177YT]

Yah sloppy derivation of physics still drives me mad. How can physicits live with all the vague heuristics and handwaving. Really crippled me and made me stick to math. /:

[@TheaHFrancis]

That professor sounds both unprofessional and unserious.

[@jose000]

.

[@jazreelangulo3796]

Yêu Đức Phúc.