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Laplace Table for differential equations
Typology: Cheat Sheet
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Name Time Domain Frequency Domain
f(t) F(s)
Impulse δ(t) 1
Step 1(t)
s
Ramp t
s^2
Monomial tn^
n!
sn+
Fractional Power ta^
Γ(a + 1)
sa+
Exponential e−at^
s + a
Sine sin(ωt)
ω
s^2 + ω^2
Cosine cos(ωt)
s
s^2 + ω^2
Decaying Sine e−σt^ sin(ωt)
ω
(s + σ)^2 + ω^2
Decaying Cosine e−σt^ cos(ωt)
s + σ
(s + σ)^2 + ω^2
Hyperbolic Sine sinh(ωt) = (^12)
eωt^ − e−ωt
) (^) ω
s^2 − ω^2
Hyperbolic Cosine cosh(ωt) = (^12)
eωt^ + e−ωt
) (^) s
s^2 − ω^2
Table 1: Laplace transforms of common functions
Property Name Property
Definition F (s) = L
f (t)
0 f^ (t)e
−stdt
Scale Homogeneity L
af (t)
= aF (s)
Super-Position L
f (t) + g(t)
= F (s) + G(s)
Linearity L
af (t) + bg(t)
= aF (s) + bG(s)
Frequency Shift L
eatf (t)
= F (s − a)
Time Shift L
f (t − T )
= e−sT^ F (s)
Time Scaling L
f (at)
s a
Derivative L
f˙ (t)
= sF (s) − f (0)
n-th Derivative L
dn
dtn^
f (t)
= snF (s) −
∑n− 1
k=
sk^
dk
dtk^
f (0)
Integral L
[∫^ t
0
f (τ )dτ
s
F (s)
Convolution L
(f ∗ g)(t)
= F (s)G(s)
Pointwise Multiplication L
f (t)g(t)
= (F ∗ G)(s)
Final Value Theorem f (∞) = limt→∞ f (t) = lims→ 0 sF (s)
Initial Value Theorem f (0) = limt→ 0 f (t) = lims→∞ sF (s)
Table 2: Properties of Laplace transforms
