Tabel Laplace [PDF]

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Differential Equations



f (t ) = L



−1



{F ( s )}



1.



1



3.



t n , n = 1, 2,3,…



F ( s ) = L { f ( t )} 1 s n! s n +1



t



3 2



7.



sin ( at )



9.



t sin ( at ) sin ( at ) − at cos ( at )



2s a 2 s + a2 2as



(s



2



+ a2 )



2



2 2



2



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,…



(s − a)



25.



uc ( t ) = u ( t − c )



t cos ( at )



(s − a)



2



+ b2



20.



e at cos ( bt )



−b



22.



e at cosh ( bt )



24.



f ( ct )



26.



δ (t − c )



b



(s − a)



2



2



n! n +1



31.



1 f (t ) t



∫



∞ s



s p +1 1 ⋅ 3 ⋅ 5⋯ ( 2n − 1) π n+ 1



2n s 2 s 2 s + a2 s2 − a2



(s



+ a2 )



2



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



2



28.



uc ( t ) g ( t )



30.



t n f ( t ) , n = 1, 2,3,…



32.



∫ f ( v ) dv



34.



sF ( s ) − f ( 0 )



36.



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



(s − a)



2



+ b2



s−a



(s − a)



2



− b2



e − cs e − cs L { g ( t + c )}



( −1)



f (t + T ) = f (t )



n



F ( n) ( s )



F ( s) s



0



f ′′ ( t )



2 2



2



t



F ( s)G ( s)



2



1 s F  c c



Dirac Delta Function



F ( u ) du



2



2as 2



sin ( at ) + at cos ( at )



cosh ( at )



F ( s − c)



© 2007 Paul Dawkins



10.



18.



ect f ( t )



f ( n) ( t )



cos ( at )



, n = 1, 2, 3,…



cos ( at + b )



29.



37.



8.



n − 12



16.



uc ( t ) f ( t − c )



f ′ (t )



t



cos ( at ) + at sin ( at )



27.



35.



6.



14.



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



Heaviside Function



0



t p , p > -1



1 s−a Γ ( p + 1)



2



2 2



2



e − cs s − cs e F ( s)



∫



4.



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



2



cos ( at ) − at sin ( at )



f ( t − τ ) g (τ ) dτ



e at



12.



(s + a ) s(s − a ) (s + a )



13.



t



2.



2a 3 2



33.



f ( t ) = L −1 {F ( s )}



π



5.



11.



Table of Laplace Transforms



∫



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 )⋯ − sf ( n − 2) ( 0 ) − f ( n−1) ( 0 ) 61



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