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Unformatted text preview: ) d d where = + = Z t Z f ( ) g ( - ) h ( t- ) d d = Z t Z f ( ) g ( - )] d h ( t- ) d = Z t ( f * g )( ) h ( t- ) d = ( f * g ) * h ( t ) . Theorem 1 (The convolution theorem) . Let f and g be piecewise con-tinuous and have exponential order . If F ( s ) = L f ( s ) and G ( s ) = L g ( s ) , then (3) F ( s ) G ( s ) = L ( f * g )( s ) for s > . A proof can be found in [1, Theorem 2.39, pp. 92-93]. References  Joel L. Schi. The Laplace Transform Theory and Applications . Springer-Verlag New York, Inc., 1999....
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