f :< 0, +") C
Ä… Ä… M > 0 t "< 0, +")
|f(t)| d" M · eÄ…t
L [f(t)] f(t) t
f(t) F (z)

+"
F (z) = L [f(t)] = f(t) · e-ztdt
0
f(t) F (z)
f(t) Ä… F (z)
Rez > Ä…

+"
(n)
F (z) = (-1)n tnf(t)e-ztdt
0
f(t) = et

+" +" +" +"
1
F (z) = f(t)e-ztdt = ete-ztdt = e(-z)tdt = ( - z)e(-z)tdt =
 - z
0 0 0 0
+"

1 1 1

e(-z)t = (0 - 1) =

 - z  - z z - 
0
L[af(t) + bg(t)] = aL[f(t)] + bL[g(t)] a, b " C

1 z
L[f(at)] = F a " R+
a a
L[f(t - a)] = e-zaF (z) a " R+

L etf(t) = F (z - )  " C
(n)
L [tnf(t)] = (-1)nF (z)

L f(n)(t) = znF (z) - zn-1f(t0) - zn-2f (t1)... - zf(n-2)(tn-2) - f(n-1)(tn-1)
f(t0), f (t1), f(n-2)(tn-2), f(n-1)(tn-1)
http : //www.mini.pw.edu.pl/ <" eplonkow/
 tn et sin t cos t tnet sin(Ét) cos(Ét) et sin(Ét)
n! 1 1 z 1 z n! É z É
zn+1 z- z2+1 z2+1 z2-1 z2-1 (z-)n+1 z2+É2 z2+É2 (z-)2+É2

t
L te-2t cos
3
z
F1(z) = L [cos t] =
z2 + 1

t 1 9z
F2(z) = L cos = F1(3z) =
3 1/3 9z2 + 1

t 9(z + 2)
F3(z) = L e-2t cos = F2 (z - (-2)) =
3 9(z + 2)2 + 1


9 9(z + 2)2 - 1
t 9(z + 2)
F4(z) = L te-2t cos = (-1)1F3 = - =
3 9(z + 2)2 + 1
(9(z + 2)2 + 1)2
f(t) g(t)

t
f(t) " g(t) = f(u)g(t - u)du
0
f(t) " g(t) = g(t) " f(t)
L [f(t) " g(t)] = F (z) · G(z)

d
L (f(t) " g(t)) = z · F (z) · G(z)
dt

t
1
L-1 F (z) = f(u)du
z
0
L-1 [F1(z)F2(z)] = f1(t) " f2(t)
d
L-1 [z · F1(z)F2(z)] = [f1(t) " f2(t)]
dt
Qm(x) An An-1 A1 Pk-1(x)
= + + ... + +
(ax - b)nPk(x) (ax - b)n (ax - b)n-1 (ax - b) Pk(x)
z
F (z) =
(z-1)(z2+1)
1
y(t) = (et - cost - sint)
2
http : //www.mini.pw.edu.pl/ <" eplonkow/
any(n) + an-1y(n - 1) + ... + a1y + a0y = f(t)
an an-1 a1 an " R an = 0

y(t)
y(0) = P0 y (0) = P1 yn-1(0) = Pn-1
L [f(t)] = F (z)
L [y(t)] = Y (z)
L [y (t)] = zY (z) - P0
L [y (t)] = z2Y (z) - zP0 - P1

L y(n)(t) = znY (z) - zn-1P0 - ..... - Pn-1
F (z) + M
Y (z) =
anzn + ...a1z + a0
y(t) y(t) = L-1[Y (z)]
y + y = 1
y(0) = y (0) = y (0) = 1
L(1) = 1/z
L (y) = Y (z)
L (y ) = zY (z) - 1
L (y ) = z2Y (z) - z - 1
L (y ) = z3Y (z) - z2 - z - 1
L(y ) + L(y ) = L(1)
1
z3Y (z) - z2 - z - 1 + zY (z) - 1 =
z

1
z3 + z Y (z) = z2 + z + 2 +
z
z3 + z2 + 2z + 1
Y (z) =
z4 + z2
2 1 -z
Y (z) = + +
z z2 z2 + 1
L-1(1/z) = 1 L-1(1/z2) = t L-1(z/z2 + 1) = cos t
y(t) = L-1 [Y (z)] = 2 · 1 + t - cos t
http : //www.mini.pw.edu.pl/ <" eplonkow/