# what are $\mathcal{O}(x)$ and $\mathcal{o}(x)$? and where did they came from?

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.

• these are so-called Landau symbols, big-o and small-o notation. see, for a start, en.wikipedia.org/wiki/Big_O_notation
– daw
Commented Mar 5 at 15:45
• @daw do you have any book recommendations ?
– pie
Commented Mar 5 at 20:43
• Why are there so many downvotes on this question? Commented Mar 7 at 18:59

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 $$f/g \to 1.$$