[texhax] "big O" and "little O" notation in amsmath
Randolph J. Herber
herber at dcdrjh.fnal.gov
Fri Jan 9 19:30:36 CET 2004
The following header lines retained to effect attribution:
>Date: Fri, 09 Jan 2004 11:09:40 -0500 (EST)
>From: Barbara Beeton <bnb at ams.org>
>Subject: Re: [texhax] "big O" and "little O" notation in amsmath
>In-reply-to:
>To: John Wheeler <jcw at chemj2.ucsd.edu>
>Cc: texhax at tug.org
>References:
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>john wheeler wrote,
> I use the amsmath package and would like to have implemented the
> "big O" and "little O" notations for specifying the behavior of a
> function in a limit. I've looked at the .sty file and done a little
> experimentation, but to no avail. Can anyone tell me if they are
> available in the package, and if so what is the usage? ...
http://primes.utm.edu/glossary/page.php?sort=BigOh
Suppose f(x) and g(x) are real valued functions defined for all x
greater than some fixed positive real x0. We write
f(x) = O(g(x)) (and we say "f(x) is big-O of g(x)"),
if there is some constant C such that
|f(x)| < C.g(x)
That is, f(x) is O(g(x)) if f is bounded by a constant times g.
For example, 53x2+23x+500 = O(x2), sin(x) = O(1), and any
polynomial in x of degree at most n is O(xn).
This big-O notation was introduced by P. Brachmann in 1894.
http://primes.utm.edu/glossary/page.php?sort=LittleOh
Suppose f(x) and g(x) are real valued functions defined for all x >
x0 (where x0 is a fixed positive real). We write
f(x) = o(g(x))
if the limit as x approaches infinity of f(x)/g(x) is zero (that
is, if eventually f(x)/g(x) becomes less than any given positive
number). Examples: 10000x = o(x2), log(x) = o(x), and xn =
o(ex). Notice that f(x) = o(g(x)) implies, and is stronger than,
f(x) = O(g(x)).
We often use the little-oh notation this way:
f(x) = g(x) + o(h(x)).
This intuitively means that the error in using g(x) to
approximate f(x) is negligible in comparison to h(x).
The little-oh notation was first used by E. Landau in 1909.
>unfortunately, i'm not familiar with the terms "big O"
>and "little O" notations. can you cite a reference
>where they are used? (the purpose of this question is
>to determine if this notation is something that should
>be supported by unicode. that has nothing to do with
>amsmath, but is related to the STIX project that ams
>has been participating in along with other scientific
>publishers; see http://www.ams.org/STIX/ .)
>they are not in the amsmath package, at least not by
>those names.
> ... Also, is a manual or other descriptive document
> available? and if so, where?
>the "amsmath user's guide" can be found at
> http://www.ams.org/tex/amslatex.html
>the book "math into latex" by george gratzer gives a
>much more detailed presentation, with lots of examples.
> -- bb
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Randolph J. Herber, herber at fnal.gov, +1 630 840 2966, CD/CDFTF PK-149F,
Mail Stop 318, Fermilab, Kirk & Pine Rds., PO Box 500, Batavia, IL 60510-0500,
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