Page 323 - Schaum's Outline of Theory and Problems of Advanced Calculus

314 IMPROPER INTEGRALS [CHAP. 12

2. Dirichlet’s test. Suppose that
(a) ðxÞ is a positive monotonic decreasing function which approaches zero as x !1.
u
ð

(b) f ðx; Þ dx < P for all u > a and 1 @ @ 2 .

a ð
1
Then the integral f ðx; Þ ðxÞ dx is uniformly convergent for 1 @ @ 2 .
a
THEOREMS ON UNIFORMLY CONVERGENT INTEGRALS
ð
1
Theorem 6. If f ðx; Þ is continuous for x A a and 1 @ @ 2 , and if f ðx; Þ dx is uniformly
ð a
1
f ðx; Þ dx is continous in 1 @ @ 2 . In particular, if
convergent for 1 @ @ 2 , then ð Þ¼
a
0 is any point of 1 @ @ 2 ,we can write
ð ð
1 1
lim ð Þ¼ lim f ðx; Þ dx ¼ lim f ðx; Þ dx ð9Þ
! 0 ! 0 ! 0
a a
If 0 is one of the end points, we use right or left hand limits.
Theorem 7. Under the conditions of Theorem 6, we can integrate ð Þ with respect to from 1 to 2 to
obtain
ð ð ð ð ð
2 2 1 1 2
f ðx; Þ d dx
ð Þ d ¼ f ðx; Þ dx d ¼ ð10Þ
1 1 a a 1
which corresponds to a change of the order of integration.
Theorem 8. If f ðx; Þ is continuous and has a continuous partial derivative with respect to for x A a
ð
1 @f
and 1 @ @ 2 , and if dx converges uniformly in 1 @ @ 2 , then if a does not depend on ,
a @
d ð 1 @f
dx
d ¼ a @ ð11Þ
If a depends on , this result is easily modified (see Leibnitz’s rule, Page 186).

EVALUATION OF DEFINITE INTEGRALS
Evaluation of definite integrals which are improper can be achieved by a variety of techniques. One
useful device consists of introducing an appropriately placed parameter in the integral and then differ-
entiating or integrating with respect to the parameter, employing the above properties of uniform
convergence.

LAPLACE TRANSFORMS

Operators that transform one set of objects into another are common in mathematics. The
derivative and the indefinite integral both are examples. Logarithms provide an immediate arithmetic
advantage by replacing multiplication, division, and powers, respectively, by the relatively simpler
processes of addition, subtraction, and multiplication. After obtaining a result with logarithms an
anti-logarithm procedure is necessary to find its image in the original framework. The Laplace trans-
form has a role similar to that of logarithms but in the more sophisticated world of differential
equations. (See Problems 12.34 and 12.36.)