Section65

# Section65 - Laplace Transform of Dirac Delta function(1 of...

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1 Laplace Transform of Dirac Delta function δ (1 of 2) The Laplace Transform of δ is defined by and thus { } { } 0 , ) ( lim ) ( 0 0 0 0 > = t t t d L t t L τ τ δ { } ( ) ( ) [ ] 0 0 0 0 0 0 0 0 0 0 ) cosh( lim ) sinh( lim 2 lim 2 1 lim 2 lim 2 1 lim ) ( lim ) ( 0 0 0 0 0 0 0 0 0 0 st st st s s st t s t s t t st t t st st e s s s e s s e e e s e e e s s e dt e dt t t d e t t L + + + = = = = + = = = = τ τ τ τ τ τ τ δ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ + < < = otherwise , 0 , 2 1 ) ( 0 0 0 τ τ τ τ t t t t t d 0 ), ( lim ) ( 0 0 0 0 > = t t t d t t τ τ δ Laplace Transform of δ (2 of 2) Thus the Laplace Transform of δ is For Laplace Transform of δ at t 0 = 0, take limit as follows: For example, when t 0 = 10, we have L { δ ( t -10)} = e - 10 s . { } 0 , ) ( 0 0 0 > = t e t t L st δ { } { } 1 lim ) ( lim ) ( 0 0 0 0 0 0 = = = st t e t t d L t L τ τ δ

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2 Product of Continuous Functions and δ
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