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I am looking for some titles. Not looking for basic textbooks nor advanced, I am craving for real stuff. More in detail I would like some book that covers calculus in one variable from a more mature perspective ( such as the one that a phd student should have). Something that may be helpful in technical situation, not a good read, not for pleasure: not the book I deserve but the one I need. It must be a huge pile of tricks and inequalities.

I must say that I am at my last year at uni as a math student, and I have read or at list I am aware of the classics such as Rudin, to make an example. They are great but not quite what I intend here. I don't need ( I hope) to learn about calculus, I need advanced tools, the kind of things that you run into by luck and you keep using ever after.

Thank you.

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  • $\begingroup$ Calculus by T. Apostal is very good! $\endgroup$ – user579462 Aug 21 '18 at 18:08
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Technical recommendations:

  1. Generatingfunctionology by Wilf and Analytic Combinatorics by Flajolet-Sedgewick. They cover the huge analogy between combinatorial classes and power series, the former with a focus on combinatorics, the latter with a focus on asymptotics (Laplace and saddle point method, Hayman conditions etc);
  2. The Cauchy-Schwarz Master Class by M.Steele and Secrets in Inequalities by Pham Kim Hung. They provide a great source of knowledge in dealing with algebraic inequalities;
  3. Advanced Integration Techniques by Zaid Alyafeai and Inside Interesting Integrals by Nahin, for learning Feynman's trick, contour integration and much more;
  4. My Superior Mathematics from an Elementary point of view (containing a bit of everything from the previous points) and, since I believe you are Italian, this unfinished book Introduzione all'Analisi Matematica, especially the part about advanced inequalities (Carleman, Hilbert, Knopp, Gagliardo-Niremberg).

The kind of things that you run into by luck and you keep using ever after, I hope.

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Rudin "Principles of Mathematical Analysis". Also known as "baby Rudin"...

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I suggest you start with Spivak's Calculus, which is meant to be an encounter with real math, and then look at books on Real Analysis, such a Rudin, Royden and the like.

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  • $\begingroup$ I disagree with Spivak's book. The book doesn't even define function properly, and many hypotheses are missing in the theorems/problems. It has great exercices though. I would suggest using another book and making the interesting exercices from this book. $\endgroup$ – user370967 Aug 21 '18 at 18:02
  • $\begingroup$ Math_QED: I totally disagree with your comment. I taught 14 times out of the book and the students loved it. If you're talking about early editions, mistakes have been caught and corrected in the third and fourth editions. If you're talking about recent editions, I'd love to hear. And the definition of function is just fine. I don't know what you're complaining about. This is not meant to be written like a graduate text. $\endgroup$ – Ted Shifrin Aug 21 '18 at 18:07
  • $\begingroup$ @TedShifrin I talk about third edition, which I own. The definition does not mention codomains, which is IMO critical for the function concept, but I guess it doesn't matter that much on this level. The limit concept could be made more general, and many times the hypothesis that the point at which the limit is taken must be a limit point or something similar is not mentioned (for example when one talks about left and right limits and the equivalence between their existence and the 'entire' limit (apply this to $x \mapsto \sqrt{x}$ at $x=0$ and the theorem as he stated fails miserably) $\endgroup$ – user370967 Aug 21 '18 at 18:14
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    $\begingroup$ When you're talking about almost all of Spivak's book, we're considering only real-valued functions, so the codomain is $\Bbb R$. We do not need to use $f\colon X\to Y$ notation in a beginning text. And your complaints are coming from wanting to do things in a totally general real analysis context. That's not the point. There's more to pedagogy than doing things in the most general setting possible. $\endgroup$ – Ted Shifrin Aug 21 '18 at 18:17
  • $\begingroup$ Moreover, I think I'm slightly biased because I read the book after I had been taught metric space analysis, which is clearly more advanced and makes you focus on these kinds of details. Maybe if you aren't aware of the details, or an instructor fills them up you can get a lot out of it. $\endgroup$ – user370967 Aug 21 '18 at 18:19
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The book "Real numbers and real analysis" by Ethan Block is really a master piece. It's single variable analysis.

It is a very rigorous book, but takes the time to explain everything and gives lots of examples (this makes the book rather lengthy). I think you can't get much more rigorous than this book.

It constructs the real numbers starting from the Peano axioms (be aware: this takes more than 100 pages ane is very very technical! But you can skip it safely if your main interest is analysis, at least if you are really familiar with suprema and infina)

It then introduces the concept of limits, which is what I believe the weaker part of the book, because it restricts the limit concept to functions defined on an interval. A more general definition of limit exists (see Rudin's book principles of mathematical analysis or Apostol's book mathematical analysis), and is worth studying.

The next section introduces derivatives and it really does a great job formalizing notions that are intuitive. Note that derivatives here are again defined as limits of functions on intervals, but this is no real problem as most authors do this (the derivative is a concept that does not make too much physical sense on weird domains).

The next section is about Riemann integration. Most analysis books give one way to get to the core theorems. This book provides more than one way, and all are equivalent.

It also introduces the Riemann-Stieltjes integral in the exercices, which is not found in many introduction books.

The next sections are about limits of functions, sequences and series and are equally great.

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For Analysis, study (i) Mathematical Analysis (Tom M. Apostol), (ii) Real and Complex Analysis (Walter Rudin), (iii) Real Analysis (Royden, Fitzpatrick). Real Analysis (N.L. Carothers) is good, too.

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  • $\begingroup$ Those books aren't about singlevariable calculus, but about general metric spaces, and is probably too advanced for OP. $\endgroup$ – user370967 Aug 21 '18 at 18:04
  • $\begingroup$ Okay... Then how about Spivak ?? $\endgroup$ – Anik Bhowmick Aug 21 '18 at 18:07
  • $\begingroup$ See my comments on another post. $\endgroup$ – user370967 Aug 21 '18 at 18:16
  • $\begingroup$ I see. I haven't studied Spivak, but I've heard about it. What can be a good book for single variable calculus ?? $\endgroup$ – Anik Bhowmick Aug 21 '18 at 18:18
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    $\begingroup$ See my answer. Another good book is Stephen Abott's : Understanding analysis. $\endgroup$ – user370967 Aug 21 '18 at 18:19
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"Introduction to measure theory" by Tao is great, it's also online

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I just loved Penner and Best's Calculus. I've in some sense been a P hd student for $28$ some odd years, (whether a good thing or a bad thing); but I just remember those $2$ volumes very fondly (used them in high school)...

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