Laplace Transforms f (s ) Lf (t ) f (t ) L1 f (s ) 1 a 1 s a 1 s a 1 e at e at t tn , 1 s a s n = a positive integer u(t ) s2 n! s n 1 1 s u(t a) e as s u(t a)f (t a) e as f (s) Pulse of β in time a 1 e as s (t ) 1 sin bt 2 cos bt sinh bt cosh bt b s b2 s s2 b2 b s2 b2 s s2 b2 e at f (t ) f (s a) f (at ) 1 s f , a a f (t) sf (s) f (0) f (n ) (t ) s n f (s ) s n 1f (0) s n 2f (0) f (n 1) (0) tf (t ) t 2 f (t ) t n f (t) t f (t)dt 0 a0 df (s) ds d 2 f (s) ds 2 (1)n f (s) s d n f (s) dsn