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Table of Laplace Transform Operations

Appendix

Table of Laplace Transform Operations

F(s) f(t)
c F (s) + c F (s) c f (t) + c f (t)
2 2
1 1
1 1
2 2
t
F(as) (a > 0) 1 f( )
a a
at
F(s − a) e f(t)
e −as F(s) (a ≥ 0) u (t)f(t − a)
a
+
sF(s) − f(0 ) f (t)
′
2
+
+
′′
s F(s) − sf(0 ) − f (0 ) f (t)
′
+
+
n
s F(s) − s n−1 f(0 ) − s n−2 ′ + (n−1) (0 ) f (n) (t)
f (0 ) − · · · − f
F(s) ∫ t f(τ)dτ
s 0
F (s) −tf(t)
′
n n
F (n) (s) (−1) t f(t)
∫
∞ F(x)dx f(t)
s t
∫ t
F(s)G(s) f(τ)f(t − τ)dτ
0
+
lim sF(s) lim f(t) = f(0 )
s→∞ t→0 +
lim sF(s) lim f(t)
s→0 t→∞
Table of Laplace Transform
F(s) f(t)

1 δ(t)
1 1
s
1 t
s 2
1 (n = 1, 2, 3, . . .) t n−1
s n (n−1)!
1 (ν > 0) t ν−1
s ν Γ(ν−1)
t
(s−1) n (n = 0, 1, 2, . . .) L (t) = e d n (t e ) Laguerre polynomials
n −t
s n+1 n n! dt n
1 e at
s−a
1 1 (e − 1)
at
s(s−a) a
at
1 (a ̸= b) e −e bt
(s−a)(s−b) a−b
at
s (a ̸= b) ae −be bt
(s−a)(s−b) a−b
s (1 + at)e at
(s−a) 2
a sin at
2
s +a 2
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