Page 322 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 12] IMPROPER INTEGRALS 313

IMPROPER INTEGRALS OF THE THIRD KIND
Improper integrals of the third kind can be expressed in terms of improper integrals of the first and
second kinds, and hence the question of their convergence or divergence is answered by using results
already established.

IMPROPER INTEGRALS CONTAINING A PARAMETER, UNIFORM CONVERGENCE
Let
ð
1
f ðx; Þ dx
ð Þ¼ ð8Þ
a
This integral is analogous to an infinite series of functions. In seeking conditions under which we
may differentiate or integrate ð Þ with respect to ,itisconvenient to introduce the concept of uniform
convergence for integrals by analogy with infinite series.
We shall suppose that the integral (8) converges for 1 @ @ 2 ,orbriefly ½ 1 ; 2 Š.
Definition.
The integral (8)is said to be uniformly convergent in ½ 1 ; 2 Š if for each > 0, we can find a number N
depending on but not on , such that
u
ð

f ðx; Þ dx <

ð Þ for all u > N and all in ½ 1 ; 2 Š

a
u
ð
ð
1
f ðx; Þ dx , which is analogous in

This can be restated by nothing that ð Þ f ðx; Þ dx ¼

a u
an infinite series to the absolute value of the remainder after N terms.
The above definition and the properties of uniform convergence to be developed are formulated in
terms of improper integrals of the first kind. However, analogous results can be given for improper
integrals of the second and third kinds.
SPECIAL TESTS FOR UNIFORM CONVERGENCE OF INTEGRALS
1. Weierstrass M test. If we can find a function MðxÞ A 0 such that
1 @ @ 2 ; x > a
(a) j f ðx; Þj @ MðxÞ
ð
1
(b) MðxÞ dx converges,
a
ð
1
then f ðx; Þ dx is uniformly and absolutely convergent in 1 @ @ 2 .
a
1 1 dx 1 cos x
ð ð

EXAMPLE. Since @ and converges, it follows that dx is uniformly
cos x
x þ 1 x þ 1 0 x þ 1 0 x þ 1
2 2 2 2
and absolutely convergent for all real values of .
As in the case of infinite series, it is possible for integrals to be uniformly convergent
without being absolutely convergent, and conversely.