Concrete mathematics : a foundation for computer science

You are the fairest
of your sex,
Let me be your
hero;
I love you as
one over x,
As x approaches
zero.
Positively.
9.2 0 NOTATION 431
But wait a minute. What if the variable n isn’t an integer? What if we
have a formula like S(x) = $x3 + ix2 + ix, where x is a real number? Then we
cannot say that S(x) = 0(x3), because the ratio S(x)/x3 = 3 + ix-’ + ix 2
becomes unbounded when x + 0. And we cannot say that S(x) = O(x),
because the ratio S(x)/x = $x2 + ix + i becomes unbounded when x t 00.
So we apparently can’t use O-notation with S(x).
The answer to this dilemma is that variables used with 0 are generally
subject to side conditions. For example, if we stipulate that 1x1 3 1, or that
x 3 c where E is any positive constant, or that x is an integer, then we can
write S(x) = 0(x3). If we stipulate that 1x1 6 1, or that 1x1 6 c where c is
any positive constant, then we can write S(x) = O(x). The O-notation is
governed by its environment, by constraints on the variables involved.
These constraints are often specified by a limiting relation. For example,
we might say that
f(n) = O(s(n)) as n-3 03. (9.13)
This means that the O-condition is supposed to hold when n is “near” co;
we don’t care what happens unless n is quite large. Moreover, we don’t
even specify exactly what “near” means; in such cases each appearance of 0
implicitly asserts the existence of two constants C and no, such that
(f(n)1 6 Clg(n)l whenever n > no. (9.14)
The values of C and no might be different for each 0, but they do not depend
on n. Similarly, the notation
f(x) = qdxl) asx+O
means that there exist two constants C and c such that
(f(x)/ 6 Clg(xl( whenever 1x1 6 E. (9.15)
The limiting value does not have to be co or 0; we can write
lnz = z-l+O((z-1)2) as 2 -3 1,
because it can be proved that Ilnz-z+l/ 6 /z- 112 when lz- 11 6 5.
Our definition of 0 has gradually developed, over a few pages, from something
that seemed pretty obvious to something that seems rather complex; we
now have 0 representing an undefined function and either one or two unspecified
constants, depending on the environment. This may seem complicated
enough for any reasonable notation, but it’s still not the whole story! Another