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IMPROPER INTEGRALS
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[CHAP. 12
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Dirichlet s test. Suppose that (a) x is a positive monotonic decreasing function which approaches zero as x ! 1.   u   (b)  f x; dx < P for all u > a and 1 @ @ 2 .   1 a Then the integral f x; x dx is uniformly convergent for 1 @ @ 2 .
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THEOREMS ON UNIFORMLY CONVERGENT INTEGRALS Theorem 6.
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1 If f x; is continuous for x A a and 1 @ @ 2 , and if f x; dx is uniformly 1 a f x; dx is continous in 1 @ @ 2 . In particular, if convergent for 1 @ @ 2 , then
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0 is any point of 1 @ @ 2 , we can write 1 1 f x; dx lim f x; dx lim  lim
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If 0 is one of the end points, we use right or left hand limits. Theorem 7. obtain Under the conditions of Theorem 6, we can integrate  with respect to from 1 to 2 to 2
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' ' 1 & 2 f x; dx d f x; d dx
a 1
10
which corresponds to a change of the order of integration. If f x; is continuous and has a continuous partial derivative with respect to for x A a 1 @f and 1 @ @ 2 , and if dx converges uniformly in 1 @ @ 2 , then if a does not depend on , @ a 1 d @f dx 11 d a @ If a depends on , this result is easily modi ed (see Leibnitz s rule, Page 186). Theorem 8.
EVALUATION OF DEFINITE INTEGRALS Evaluation of de nite 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 di erentiating 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 inde nite 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 nd its image in the original framework. The Laplace transform has a role similar to that of logarithms but in the more sophisticated world of di erential equations. (See Problems 12.34 and 12.36.)
CHAP. 12]
IMPROPER INTEGRALS
The Laplace transform of a function F x is de ned as 1 f s lfF x g e sx F x dx 12
F x a eax sin ax cos ax xn n 1; 2; 3; . . . a 8
lfF x g 8>0 8>a 8>0 8>0 8>0
and is analogous to power series as seen by replacing e s by t so that e sx tx . Many properties of power series also apply to Laplace transforms. The adjacent short table of Laplace transforms is useful. In each case a is a real constant.
1 8 a a 82 a2 8 82 a2 n! 8n 1
LINEARITY The Laplace transform is a linear operator, i.e., fF x G x g fF x g fG x g:
Y 0 x Y 00 x
8lfY x g Y 0 82 lfY x g 8Y 0 Y 0 0
This property is essential for returning to the solution after having calculated in the setting of the transforms. (See the following example and the previously cited problems.)
CONVERGENCE The exponential e st contributes to the convergence of the improper integral. What is required is that F x does not approach in nity too rapidly as x ! 1. This is formally stated as follows: If there is some constant a such that jF x j eax for all su ciently large values of x, then
f s
e sx F x dx converges when s > a and f has derivatives of all orders.
(The di erentiations
of f can occur under the integral sign >.)
APPLICATION The feature of the Laplace transform that (when combined with linearity) establishes it as a tool for solving di erential equations is revealed by applying integration by parts to f s letting u F t and dv e st dt, we obtain after letting x ! 1 x 1 1 1 st 0 e st F t dt F 0 e F t dt: s s 0 0