1. f (t ) Laplace{ f (t )} Trasformata Z
h( t ) 1 z
s z −1
1 Tz
t
s2 ( z − 1)2
t2 1 T 2 z( z + 1)
2 s3 2( z − 1)3
k −1
( k −1)!
k −1 ∂ k −1 z #
lim ( −1)
! $
t
sk a→ 0 ∂a k −1 z − e − aT
e − at 1 z
s+a z − e − aT
te − at 1 Tze − aT
( s + a )2 ( z − e − aT )2

2. t k e − at k! ∂k z #
! $
k
( −1)
(s + a)k +1 ∂a k z − e − aT
1− e − at a z(1 − e − aT )
s( s + a ) ( z − 1)( z − e − aT )
1− e − at a z[( aT − 1 + e − aT )z + (1 − e − aT − aTe − aT ) ]
t−
a s2 ( s + a) a( z − 1)2 ( z − e − aT )
− at a2 z z aTe − aT z
1 − (1 + at )e − − aT −
s ( s + a )2 z −1 z − e ( z − e − aT )2
b−a (e − aT − e − bT )z
e − at − e − bt
(s + b)(s + a ) ( z − e − aT )( z − e − bT )
sin( at ) a z sin( aT )
s2 + a2 z 2 − 2 z cos( aT ) + 1
s z( z − cos( aT ))
cos( at )
s2 + a2 z 2 − 2 z cos( aT ) + 1

3. − at b ze − aT sin( bT )
e sin( bt )
( s + a )2 + b 2 z 2 − 2 ze − aT cos(bT ) + e −2 aT
e − at cos( bt ) s+a z 2 − ze − aT cos(bT )
( s + a )2 + b 2 z 2 − 2 ze − aT cos(bT ) + e −2 aT
( Az + B)z
1 − e − at (cos(bt ) + b2 + a2 ( z − 1)( z 2 − 2 ze − aT cos(bT ) + e −2 aT )
a s ( s + a)2 + b 2
+ sin(bt )) a
A = 1 − e − aT (cos( bT ) + sin( bT ) )
b b
a
B = e −2 aT + e − aT ( sin( bT ) − cos(bT ))
b
1 e − at ( Az + B) z
+ + 1
ab a( a − b )
s( s + a)( s + b) ( z − 1)( z − e − aT ) ( z − e − bT )
e − bt
+
b( b − a )