Tableau de transformée de laplace1. FORMULA FOR LAPLACE TRANSFORM and its inverses
S.NO f(t) L[f(t)] L-1 (L[f(t)])= f(t)
1 1 L[1] = 1/s 1/s =1
2 t L[t] = 1/s2 1/s2 = t
3 tn L[tn] = n!/sn+1 1/sn = tn-1/(n-1)!
4 tn L[tn] = Ғ(n+1)/sn+1,n-
non integer
5. 𝑒 𝑎𝑡
L[𝑒 𝑎𝑡
] = 1/s-a 1/s-a = 𝑒 𝑎𝑡
6 𝑒−𝑎𝑡
L[𝑒−𝑎𝑡
] = 1/s+a 1/s+a = 𝑒−𝑎𝑡
7 Sinat L[Sinat] = a/𝑠2
+ 𝑎2
a/𝑠2
+ 𝑎2
= 𝑠𝑖𝑛𝑎𝑡,
1/𝑠2
+ 𝑎2
= sinat/a
8 Cosat L[Cosat] == s/𝑠2
+ 𝑎2
s/𝑠2
+ 𝑎2
= Cosat
9 Sinhat L[Sinhat] == a/𝑠2
−
𝑎2
a/𝑠2
− 𝑎2
= sinhat
10 Coshat L[Coshat] == s/𝑠2
−
𝑎2
s/𝑠2
− 𝑎2
= coshat
2. Properties of LAPLACE TRANSFORM
S.NO PROPERTY f(t) L[f(t)]
1 scale f(at) 1/a F(s/a)
2 derivative 𝑓′
(t) sL[f(t)]-f(0)
𝑓′′
(𝑡) 𝑠2
𝐿[ 𝑓( 𝑡)] − 𝑠𝑓(0) −
𝑓′
(0)
3 Division by t 1/t f(t)
∫ 𝐿[ 𝑓( 𝑡)] 𝑑𝑠
∞
𝑠
4 Multiple by t tn f(t) (-1)n dn/dsn L[f(t)]
5 Initial value
theorem
lim
𝑡→0
𝑓(𝑡) lim
𝑠→∞
𝑠𝐹(𝑠)
6 Final value
theorem
lim
𝑡→∞
𝑓(𝑡) lim
𝑠→0
𝑠𝐹(𝑠)
7 First shifting
theorem
𝑒 𝑎𝑡
𝑓(𝑡) F[s-a]
𝑒−𝑎𝑡
𝑓(𝑡) F[s+a]
3. Properties ofinverses LAPLACETRANSFORM
S.NO PROPERTY
1 First shifting L-1[F(s+a)] 𝒆−𝒂𝒕
𝑳−𝟏
[F(s)]
2 derivatives L-1[𝑭′
(𝒔)] (identification :
: 𝒔+𝒂𝒏𝒚 𝒕𝒆𝒓𝒎
(𝒒𝒖𝒂𝒅𝒓𝒂𝒕𝒊𝒄 𝒆𝒒.,) 𝟐
-t L-1 F(s)
3 Division by s L-1[
𝑭(𝒔)
𝒔
] (identification :
: 𝒂𝒏𝒚 𝒕𝒆𝒓𝒎
𝒔(𝒐𝒏𝒕 𝒕𝒆𝒓𝒎)
∫ 𝐿−1
[𝐹( 𝑠)]
𝑡
0
𝑑𝑡
4 Multiple by s L-1[s F(s)] (identification :
: 𝒔
𝒒𝒖𝒂𝒅𝒓𝒂𝒕𝒊𝒄 𝒆𝒒.,
d/dt L-1F(s)
5 L-1[Log, cot ,tan functions] -1/t L-1 𝑑
𝑑𝑠
[F(s)]
6 Convolution
theorem
L[(f*g)] = L[f(t)]L[g(t)]