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m2k_opm_lapspr - Specic Properties of the Laplace Transform...

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Specific Properties of the Laplace Transform General properties of the Laplace Transform, such as linearity , i.e., ( L c 1 f 1 + c 2 f 2 ) ( s ) = c 1 ( L f 1 ) ( s ) + c 2 ( L f 2 ) ( s ), are very important but they are not specific to the transform because they are shared by many other mathematical operations. Here we will list some of the most important specific properties of the Laplace transform and show how they can be used to generate more examples of Laplace transforms of functions important in a variety of applications, including the solution of differential and integral equations. One very specific property of the transform, its behavior with respect to convolution , is so important that it is treated in a separate section. Property I: Laplace Transform of g ( x ) = e ax f ( x ). If ( L f ) ( s ) = ˆ f ( s ) is known then we can compute ( L g ) ( s ) ( L e ax f ( x )) ( s ) = ˆ f ( s - a ) . In words: the Laplace transform of e ax f ( x ) is obtained by substituting s - a for s in the formula for the Laplace transform of f . (Note that we usually omit the argument x , as in ( L f ) ( s ), but we include it, as in ( L e ax f ( x )) ( s ), if it is needed to express a particular formula.) This property is immediate from the computation ˆ g ( s ) = Z 0 e - sx e ax f ( x ) dx = Z 0 e - ( s - a ) x f ( x ) dx = ˆ f ( s - a ) . Example I(a) Since the Laplace transforms of cos bx and sin bx are s s 2 + b 2 and b s 2 + b 2 , respectively, the Laplace transforms of e ax cos bx and e ax sin bx must be s - a ( s - a ) 2 + b 2 and b ( s - a ) 2 + b 2 , respectively. Example I(b) For f ( x ) x , which has Laplace transform 1 s 2 , Prop- erty I shows that ( L e ax x ) ( s ) = 1 ( s - a ) 2 . 1
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Property II: Laplace Transform of g ( x ) x n f ( x ) We differentiate ( L f ) ( s ) = R 0 e - sx f ( x ) dx with respect to s : d ds ( L f ) ( s ) = - Z 0 e - sx x f ( x ) dx which we can also write as ( L x f ( x )) ( s ) = - d ds ( L f ) ( s ) . This relationship can be used to compute the Laplace transform of x f ( x ) when the Laplace transform of f ( x ) is already known. Example II(a) ( L 1) ( s ) = 1 s ( L x · 1) ( s ) = - 1 s 2 = 1 s 2 . Example II(b) Since ( L e ax ) ( s ) = 1 s - a , we have ( L x e ax ) ( s ) = - d ds 1 s - a = 1 ( s - a ) 2 .
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