ABSTRACT
Improper integrals are such that their range or integrands are unbounded. They are defined as the limits of certain proper integrals. Thus, an improper integral over the interval [0,∞) is defined as ∫∞
0 f(x) dx = lim
0 f(x) dx whenever the limit
exists. Improper integrals over [a,∞), (−∞, a] are similarly defined. Improper integrals over the interval (−∞,∞) are defined in two ways: First, there is the usual definition:
∫∞ −∞ f(x) dx =
∫ 0
−∞ f(x) dx + ∫∞ 0
f(x) dx. Then, there is the other definition:
∫∞ −∞ f(x) dx = limR→∞
∫ R
−R f(x) dx, provided both limits exist. This definition is also known as Cauchy’s principal value (or p.v.) of the integral, denoted by−
∫ ∞
−∞ f(x) dx. A common p.v. integral is the Hilbert transform−
∫ b
f(t) t− x dt, where
−∞ ≤ a < b ≤ ∞, and a < x < b. A sufficient condition for the Hilbert transform to exist over a finite interval [a, b] is that f(t) satisfy a Lipschitz or Ho¨lder condition in [a, b]; i.e., there are constants A > 0 and 0 < α ≤ 1 such that for any two points t1, t2 ∈ [a, b] we have
∣
∣f (t1)− f (t2) ∣
∣ ≤ A ∣∣t1 − t2 ∣
∣
α . Cauchy’s p.v. integrals are
discussed in Chapter 6.
