xiv Notation
We write lcm[a1,a2,...,ak] to denote the least common multiple of the
positive integers a1,a2,...,ak. Similarly, we write gcd(a1,a2,...,ak) to de-
note the greatest common divisor of the positive integers a1,a2,...,ak; when
the context is clear, we may at times simply write (a1,a2,...,ak) instead of
gcd(a1,a2,...,ak), as well as [a1,a2,...,ak] instead of lcm[a1,a2,...,ak].
Throughout this book, log x stands for the natural logarithm of x. At
times, we also write log2 x instead of log log x and more generally, for each
integer k 3, we let logk x stand for log(logk−1 x).
For each integer k 0, we denote by
Ck[a, b] the set of functions whose
kth derivative exists and is continuous on the interval [a, b]. Thus
C0[a,
b]
stands for the set of continuous functions on [a, b].
It is often convenient to use the notations introduced by Landau, namely
o(...) and O(...), to compare the order of magnitude of functions in the
neighborhood of a point or of infinity. Unless we indicate otherwise, we
shall mean the latter.
Given two functions f and g defined on [a, ∞), where a 0 and g(x) 0,
we shall write that f(x) = O(g(x)) if there exist two constants M 0 and
x0 such that |f(x)| Mg(x) for all x x0. In particular, f(x) = O(1) if
f(x) is a bounded function. Moreover, instead of writing f(x) = O(g(x)),
we shall at times write f(x) g(x).
Thus, we have, as x ∞,
x =
O(x2),
sin x = O(1), log x =
O(x1/10),
sin x
x
= O(1),
x4
=
O(ex),
ex
sin x =
O(ex),
2x2
+
x
3
x2,
x3
x3+x2
+ 7 = O(1).
Given two functions f and g defined on [a, ∞), where a 0 and g(x) 0,
we shall write f(x) = o(g(x)) as x if, for each ε 0, there exists a
constant x0 = x0(ε) such that |f(x)| εg(x) for all x x0.
Thus, we have
1
x
= o(1), sin x = o(x), log x = o(x),
sin 1/x
1/x
= 1 + o(1),
x4
=
o(ex), xex
sin x =
o(x2ex).
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