Definition talk:Big-O Notation

The definition of big O is different. It does not use limits, and certainly doesn't suppose that such a limit exists.--barto (talk) 18:28, 3 December 2016 (EST)


 * If you have a reliable source, you should add a separate definition according to the one in your source. Then, I would recommend adding a page showing that they are equivalent, or showing a counterexample if they're not equivalent. --GFauxPas (talk) 20:42, 3 December 2016 (EST)


 * Actually, if the definition you're referring to is the one used in the "implied constant", I think it would be a good idea to extract it and put it as a second definition. And then someone should prove their equivalence or non-. --GFauxPas (talk) 21:12, 3 December 2016 (EST)


 * There's a definition using limits on Wikipedia, but it lacks cohesion.


 * I remember that goes into considerable detail on the subject, but it's right at the bottom of a massive pile of all sorts of books and I wasn't planning on getting it out today.


 * This page was originally written, in substantially the same format, back in 2009 before we started investing effort into providing citations. The writer was a PhD student, I believe, so I was prepared to take his word for it. --prime mover (talk) 02:19, 4 December 2016 (EST)


 * It is possible that in computer science they use the simplified definition using limits, because they only consider some simple functions there, such as $n^\alpha(\log n)^\beta$. In mathematics, we do not restrain ourselves to such simple cases and Big-O is defined otherwise. This is why I'd rather not base any definitions in asymptotic analysis ($O$, $o$, $\asymp$, $\sim$, $\ll$ are the most common, as well as their non-uniform versions $O_\alpha$, $o_\alpha$, etc) on what can be found in books on computer science or (complexity of) algorithms. To convince you, the most common version of the prime number theorem is stated as $\pi(x)-x/\log x=O(x/\log^2x)$, but this does not imply that their quotient has a nonnegative limit! In asymptotic analysis we really don't care about the limit.


 * What I propose: Include the proper definition (with "implied constants", as you call it) and mention the computer-science definition in a remark. --barto (talk) 13:36, 24 January 2017 (EST)