# Properties differentiation theorem df t sf s

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Unformatted text preview: + βF2 (s) Example: y( t ) = 2a ( t ) + 3b( t ) Y(s) = ? Properties Differentiation Theorem df ( t ) = sF ( s ) − f (0) dt d 2 f (t) = s 2 F ( s ) − sf (0) − f (0) 2 dt n d n f (t) n = s F ( s ) − ∑ s n − k f k −1 (0) n k =1 dt Properties E xa m p le : dy (t ) + 3 y (t ) = 0 dt y (0) = 3 d 2 y (t ) dy (t ) + 12 + 32 y (t ) = 32u (t ) 2 dt dt y (0) = y ' (0) = 0 Properties Fina l Va lue T h e o re m (FVT ) lim f ( t ) = lim sF(s) t →∞ s →0 This theorem applies, if and only if, all poles of sF(s) lies in the left half s plane. Example. Find f(∞ ) for the following system 1 F (s) = s ( s + 1) Inverse Laplace Transform The Inverse Laplace transformation for a function F(s) is σ + j∞ 1 st f (t ) = F ( s )e ds ∫j∞ 2πj σ − However, the inversion integral is complicated. A convenient method for obtaining inverse Laplace transform is to use a table of Laplace transform. 3 f (t ) = ? F ( s) = s+4 Inverse Laplace Transform Partial fraction expansion method can be used to find the inver...
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