Is there a comprehensive calculus or analysis textbook or problem book, written in the last twenty years, that emphasizes the use of Landau notation (big and little oh), especially for making estimates and calculating limits?

This emphasis is lacking in most conventional treatments and standard undergraduate coursework, at least in my experience. (I am in the US.) Yet as so many MSE answers illustrate, it is the way many experts approach elementary problems. I feel I have learned more about the power of this approach by reading answers here at MSE than I have from any book. So I'm curious whether some systematic but elementary treatment does exist that I've just happened to miss. If not, where would you point students looking to develop facility in calculation from this point of view?

EDIT: well, since there have been no answers, I wonder if there is such a book at all, not necessarily written within the last twenty years. I suppose one reference would be Whittaker and Watson's Course of Modern Analysis. Is there something pitched at a slightly more elementary level, that would be suitable to assign to first or second year undergraduates?

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    $\begingroup$ (To be fair, many books mention or allude to it in passing, or perhaps relegate it to a small number of exercises. This seems to me to grossly misrepresent how central it is to the way working mathematicians think about limits, series expansions, etc.) $\endgroup$ Commented Jun 4, 2016 at 19:07
  • $\begingroup$ Perhaps point them to this site, and hope they learn here rather than just having their questions answered ... $\endgroup$ Commented Jun 4, 2016 at 19:31
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    $\begingroup$ It would be nice if there was some way to get a master list of nice answers which use the Landau notation powerfully. $\endgroup$ Commented Jun 4, 2016 at 20:03
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    $\begingroup$ @JamesS.Cook Looking at Olivier Oloa's list of answers may be a good start. $\endgroup$
    – Clement C.
    Commented Jun 4, 2016 at 20:18
  • $\begingroup$ @symplectomorphic If you think a bounty may help, let me know -- I am bounty-inclined. $\endgroup$
    – Clement C.
    Commented Jun 7, 2016 at 0:30

2 Answers 2


It may be difficult to find exactly what you are looking for. However, Landau notation is commonly used in books on perturbation theory, and this may be a viable alternative. I would recommend two books on perturbation theory that frequently use Landau notation:

  1. Hinch, Perturbation methods, 1991

  2. Simmonds and Mann, A First Look at Perturbation Theory, 1986

There is also a chapter on asymptotics in Keener, Principles of Applied Mathematics.

Finally, Donald Knuth wrote a nice article on the history and use of asymptotic notations: Knuth, Big omicron and big omega and big theta, 1976.


You might be better off reading a book on analytic number theory if you want exposure to this notation. In which case, I'd recommend something like Tom Apostol's "Introduction to Analytic Number Theory". A lot of people aren't a fan of this book, but I'm rather fond of it, and it's a fairly self-contained introduction to the subject (IIRC, you just need to be comfortable with basic calculus). If you're a first/second year undergraduate, it's certainly readable, but some of the later bits may be a little tricky.

From chapter 3 onwards, there's plenty of uses of Big-O notation in theorems, their proofs, and the numerous exercises. Once you're happy with Big-O, Little-o is fairly easy to wrap your head around.

  • $\begingroup$ Isn't there a difference in the definitions between the analysis and the number theoretic notations, though? Specifically for $\Omega(\cdot)$. $\endgroup$
    – Clement C.
    Commented Jun 14, 2016 at 0:44
  • $\begingroup$ @ClementC So there are different definitions of $\Omega$, but they only tend to differ from the number theoretic (á la Hardy and Littlewood) definition in computer science and computational complexity theory. In analysis, the number theoretic definition is generally used. $\endgroup$
    – MadMonty
    Commented Jun 14, 2016 at 1:14

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