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I often see $\mathcal{O}(x)$ or $o(x)$ notations in evaluating limits, sums and integrals like in this answer on one of my questions, but I have no idea what they mean. I thought that these notations and their meaning are taught in real analysis books but after reading 2 of them I can say I was wrong.

Since I am a self learner and I don't what courses should I take or not for pure math I have no Idea from which field(area) these notations came from.

So I want to specifically ask for books that deals with them (It is more about finding out what is that branch (area) of mathematics that these notations came from than finding out what these notations mean. because I think that there are more useful thing in that branch(area) or books) It would be better if there book(s) are only pure math books.

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    $\begingroup$ these are so-called Landau symbols, big-o and small-o notation. see, for a start, en.wikipedia.org/wiki/Big_O_notation $\endgroup$
    – daw
    Commented Mar 5 at 15:45
  • $\begingroup$ @daw do you have any book recommendations ? $\endgroup$
    – pie
    Commented Mar 5 at 20:43
  • $\begingroup$ Please do not answer questions in the comments. If you have an answer, please use the "Answer" dialog to post it as an answer. $\endgroup$
    – Xander Henderson ♦
    Commented Mar 5 at 20:49
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    $\begingroup$ Why are there so many downvotes on this question? $\endgroup$
    – Benjamin Wang
    Commented Mar 7 at 18:59

1 Answer 1

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Landau introduced little-oh notation in Handbuch der Lehre von der Verteilung der Primzahlen (1909) while trying to provide a similar concept to the big-oh notation from Bachman who introduced it in Analytische Zahlentheorie (1894). It is also worth mentioning Hardy's work too: Order's of infinity (1910); his notation is superior to Landau's but by the time Hardy introduced his, Landau's notation was already commonplace and Hardy's notation never got the attention it deserved.

A more modern reference would be Dieudonne's Infinitesimal Calculus, Hermann, Paris (1971), specifically Ch 3. "Asymptotic Developments." The whole book is excellent as it deals with many non-orthodox topics of calculus.

In a nutshell, $f = o(g)$ means that $|f/g| \to 0$ whereas $f = O(g)$ means $|f/g|$ is bounded near the point of approximation. Intuitively, these are functional limit versions of $<$ and $\leq,$ respectively. There are also version of $\approx$ and $=,$ namely $f = \Theta(g)$ which means, obviously $f = O(g)$ and $g = O(f),$ and $f \sim g$ which means $f/g \to 1.$

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  • $\begingroup$ Is there any more modern books ? reading a century old book can be very challenging in most cases. $\endgroup$
    – pie
    Commented Mar 5 at 21:27
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    $\begingroup$ @pie I don't expect my references to be accepted as "the right answer" but as Xander mentioned, I shouldn't give you answers (i.e., references) in the comments. I personally wrote a book in vector calculus (in normed spaces) and I extensively used the little-oh notation, specially when dealing with Taylor's theorem. Given the definition, it is easy to derive most basic properties of it. $\endgroup$
    – William M.
    Commented Mar 5 at 21:29
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    $\begingroup$ @pie Hardy's book seem to have been updated here: subdude-site.com/WebPages_Local/RefInfo/eDocs/Math_edocs/docs/… $\endgroup$
    – William M.
    Commented Mar 5 at 21:37

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