[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
>To: John Wheeler <jcw at chemj2.ucsd.edu>
>Cc: texhax at tug.org
>	<mailto:texhax-request at tug.org?subject=subscribe>
>	<mailto:texhax-request at tug.org?subject=unsubscribe>

>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? ...


    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.


    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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