ImproperIntegral

ImproperIntegral - Improper Integral Since the concept of...

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Improper Integral Since the concept of Laplace transform involves an integral from zero to infinity, the knowledge of improper integral is needed. Definition : An improper integral over an unbounded interval is defined as a limit of integrals over finite intervals; thus f(t)dt a ! " = lim A #! f(t)dt a A " where A is a positive real number. If the integral from a to A exists for each A > a, and if the limit as A exists, then the improper integral is said to converge to that limiting value. Otherwise the integral is said to diverge , or fail to exist, Examples : 1) Let f(t) = e ct , t 0 and c is a real nonzero constant. e ct dt = lim A !" e ct dt = lim A !" 0 A # 0 " # e ct c 0 A = lim A !" 1 c e cA $ 1 ( ) if c < 0 , e cA 0 as A , then the improper integral converges to -1/c, if c > 0, e cA as A , then the improper integral diverges, if c = 0 , e ct = 1, then the improper integral diverges. 2) Let f(t) = 1/t, t
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This note was uploaded on 07/23/2011 for the course MAP 2302 taught by Professor Staff during the Fall '08 term at FIU.

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ImproperIntegral - Improper Integral Since the concept of...

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