Singularities/Residues/ Applications 687- Improper integrals of mixed type. If the interval of integration is
infinite and the integrand is unbounded at a finite number of points of that
interval, then the integral is an improper integral of mixed type. In this
case each infinite discontinuity of the function and each infinite limit of
integration requires a separate limit process as indicated above in cases 1
and 2.
Many important improper integrals that arise in practice are of mixed
type. For instance, the r-integral, discussed in Section 8.20, is of mixed
type if Re z < 1. Some elementary improper integrals can be explicitly
evaluated by means of the fundamental theorem of calculus or by a number
of special methods, as shown in calculus textbooks. A more systematic and
efficient approach is provided by the method of residues.
In some cases the actual evaluation of an improper integral is not needed
but a determination of whether the integral converges or diverges. Tests
paralleling those for the convergence or divergence of series do exist. For
later use we shall state only the so-called comparison tests.
Theorem 9.11 Suppose that J(x) and g(x) are continuous for x ~a, If
0 $ l/(x)I $ g(x), and la^00 g(x) dx converges, then la^00 f(x) dx converges
andOn the other hand, if 0 $ g( x) $ f ( x) and la
00
g( x) dx diverges, then
J.(^00) f(x)dx diverges. The choice g(x) = 1/xP is often useful, since it is
a oo
known that 11 dx / xP converges for p > 1 and diverges for p $ 1.
Theorem 9.12 Suppose that f(x) and g(x) are continuous on (a, b] yet
unbounded on [a,b]. If 0 $ lf(x)l 5 g(x) and 1: g(x)dx converges, then
1: f ( x) dx converges, and
If 0 $ g(x) $ f(x) and 1: g(x) dx diverges, then 1: J(x) dx diverges also.
The choice g(x) = 1/(x - a)P is often convenient, since it is known that
1: dx/(x - a)P converges for p < 1 and diverges for p ~ 1.
We now proceed to the evaluation of some real improper integrals. To
evaluate an integral of either of the forms1
00