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 log 2 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).
Previous Page Next Page