10 - MATH2860U: Chapter 7 cont. 1 THE LAPLACE TRANSFORM...

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MATH2860U: Chapter 7 cont… 1 THE LAPLACE TRANSFORM cont… Operational Properties II (Section 7.4, pg. 282) Recall: Last day, we introduced several useful properties related to the Laplace transform. Now let’s introduce a few more. Derivatives of a Transform Theorem 7.4.1: If   ) ( ) ( t f s F L and , 3 , 2 , 1 n , then ) ( ) 1 ( )} ( { s F ds d t f t n n n n . Proof ( 1 n case): Example: Example: 2 2 1 ) 9 ( s s
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MATH2860U: Chapter 7 cont… 2 Example: Transforms of Integrals Definition: If functions f and g are piecewise continuous on ) , 0 [ , then a special product, denoted by g f , is defined by the integral t d t g f g f 0 ) ( ) ( and is called the convolution of f and g . Theorem 7.4.2 ( Convolution Theorem ): If ) ( t f and ) ( t g are piecewise continuous on ) , 0 [ and of exponential order, then      ) ( ) ( ) ( ) ( s G s F t g t f g f L Example: Example: ) 4 ( 1 2 1 s s
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10 - MATH2860U: Chapter 7 cont. 1 THE LAPLACE TRANSFORM...

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