Tabel Laplace [PDF]

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f (t ) = L



-1



{F ( s )}



1.



1



3.



t n , n = 1, 2,3,K



5.



1 s n! s n +1



p



t



3 2



7.



sin ( at )



9.



t sin ( at )



11.



Table of Laplace Transforms F ( s ) = L { f ( t )} f ( t ) = L -1 { F ( s )}



sin ( at ) - at cos ( at )



2s a 2 s + a2 2as



(s



2



+ a2 )



2a 3



2 2



2



cos ( at ) - at sin ( at )



15.



sin ( at + b )



17.



sinh ( at )



19.



e at sin ( bt )



21.



e at sinh ( bt )



23.



t ne at , n = 1, 2, 3, K



25.



uc ( t ) = u ( t - c )



2 2



(s - a)



2



+ b2



b



(s - a)



2



-b



(s - a)



n +1



27.



uc ( t ) f ( t - c )



e - cs s - cs e F (s)



29.



ect f ( t )



F (s - c)



31.



1 f (t ) t t



f ( t - t ) g (t ) dt



33.



ò



35.



f ¢(t )



37.



f ( n) ( t )



0



© 2004 Paul Dawkins



2



n!



¥ s



t p , p > -1



6.



t



8.



cos ( at )



10.



t cos ( at )



n - 12



, n = 1, 2,3,K



n+ 1



2n s 2 s 2 s + a2 s2 - a2



(s



+ a2 )



2



sin ( at ) + at cos ( at )



F ( u ) du



14.



cos ( at ) + at sin ( at )



16.



cos ( at + b )



18.



cosh ( at )



20.



e at cos ( bt )



22.



e at cosh ( bt )



24.



f ( ct )



26.



d (t - c ) Dirac Delta Function



28.



uc ( t ) g ( t )



30.



t n f ( t ) , n = 1, 2,3,K



32.



ò



t 0



f ( v ) dv



F (s)G ( s)



34.



f ( t + T ) = f (t )



sF ( s ) - f ( 0 )



36.



f ¢¢ ( t )



(s + a ) s ( s + 3a ) (s + a ) 2 2



2



2



2 2



2



s cos ( b ) - a sin ( b ) s2 + a2 s 2 s - a2 s-a



(s - a)



2



+ b2



s-a



(s - a)



2



- b2



1 æsö Fç ÷ c ècø e - cs e - cs L { g ( t + c )}



( -1)n F ( n) ( s ) F (s) s



ò



T 0



e- st f ( t ) dt



1 - e - sT s 2 F ( s ) - sf ( 0 ) - f ¢ ( 0 )



s n F ( s ) - s n-1 f ( 0 ) - s n- 2 f ¢ ( 0 ) L - sf ( n -2) ( 0 ) - f ( n-1) ( 0 ) 1



2



2as 2 2



s sin ( b ) + a cos ( b ) s2 + a2 a 2 s - a2 b



ò



4.



1 s-a G ( p + 1) s p +1 1 × 3 × 5L ( 2n - 1) p



2



2



Heaviside Function



e at



12.



(s + a ) s (s - a ) (s + a ) 2



13.



2



2.



F ( s ) = L { f ( t )}



Table Notes 1. This list is not inclusive and only contains some of the more commonly used Laplace transforms and formulas. 2. Recall the definition of hyperbolic trig functions. et + e - t et - e- t cosh ( t ) = sinh ( t ) = 2 2 3. Be careful when using “normal” trig function vs. hyperbolic trig functions. The only difference in the formulas is the “+ a2” for the “normal” trig functions becomes a “- a2” for the hyperbolic trig functions! 4. Formula #4 uses the Gamma function which is defined as ¥



G ( t ) = ò e - x xt -1 dx 0



If n is a positive integer then,



G ( n + 1) = n !



The Gamma function is an extension of the normal factorial function. Here are a couple of quick facts for the Gamma function G ( p + 1) = pG ( p ) p ( p + 1)( p + 2 )L ( p + n - 1) = æ1ö Gç ÷ = p è2ø



© 2004 Paul Dawkins



2



G ( p + n) G( p)