Laplace Transform Table
1
f (t) F (s)= L[f ](s)= Z 1 0- e -st f (t)dt u(t) 1 s , Re{s} > 0 tu(t) 1 s 2 , Re{s} > 0 t n u(t), n ≥ 0 integer n! s n+1 , Re{s} > 0 t a u(t), a ≥-1 Γ(a + 1) s a+1 , Re{s} > 0 e at u(t) 1 s - a , Re{s} > Re{a} cos(! 0 t)u(t) s s 2 + ! 2 0 , Re{s} > 0 sin(! 0 t)u(t) ! 0 s 2 + ! 2 0 , Re{s} > 0 cosh(kt)u(t) s s 2 - k 2 , Re{s} > |k| sinh(kt)u(t) k s 2 - k 2 , Re{s} > |k| u(t - a) e -as s , Re{s} > 0 d dt f (t) sF (s) - f (0-) d n dt n f (t) s n F (s) - s n-1 f (0-) - s n-2 f 0 (0-) - ··· ··· - sf (n-2) (0-) - f (n-1) (0-) R t 0 f (⌧ )d⌧ 1 s F (s) e at f (t) F (s - a) f (t) ⇤ g(t)= R 1 -1 f (⌧ )g(t - ⌧ )d⌧ F (s)G(s) t n f (t), n =1, 2,... (-1) n d n ds n F (s) f (t) t R 1 s F (σ)dσ u(t - a)f (t - a), a ≥ 0 e -as F (s) d(t - a) e -as f (t) periodic of period T , f (t) = 0, t< 0 1 1 - e -sT Z T 0 e -st f (t)dt Table 1: Basic Laplace Transform Pairs
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Electrical Engineering table for Laplace Transforms.
Transcript of Laplace Transform Table
f(t) F (s) = L[f ](s) =
Z 1
0�e
�stf(t) dt
u(t)1
s, Re{s} > 0
tu(t)1
s2, Re{s} > 0
tnu(t), n � 0 integer
n!
sn+1, Re{s} > 0
tau(t), a � �1
�(a+ 1)
sa+1, Re{s} > 0
e
atu(t)
1
s� a, Re{s} > Re{a}
cos(!0t)u(t)s
s2 + !20
, Re{s} > 0
sin(!0t)u(t)!0
s2 + !20
, Re{s} > 0
cosh(kt)u(t)s
s2 � k2, Re{s} > |k|
sinh(kt)u(t)k
s2 � k2, Re{s} > |k|
u(t� a)e
�as
s, Re{s} > 0
ddtf(t) sF (s)� f(0�)
dn
dtnf(t) snF (s)� sn�1f(0�)� sn�2f 0(0�)� · · ·
· · ·� sf (n�2)(0�)� f (n�1)
(0�)
R t
0 f(⌧) d⌧1
sF (s)
e
atf(t) F (s� a)
f(t) ⇤ g(t) =R1�1 f(⌧)g(t� ⌧) d⌧ F (s)G(s)
tnf(t), n = 1, 2, . . . (�1)
n dn
dsnF (s)
f(t)
t
R1s F (�) d�
u(t� a)f(t� a), a � 0 e
�asF (s)
d(t� a) e
�as
f(t) periodic of period T , f(t) = 0, t < 0
1
1� e
�sT
Z T
0
e
�stf(t) dt
Table 1: Basic Laplace Transform Pairs
