# Equations Laplace Transform Charts! by MrRail

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```									 Math 308: Diﬀerential Equations                                      Laplace Transform Charts!

For the entirety of this document, f, g, h are functions that have well deﬁned Laplace trans-
forms, a, b, c are constants, and L{f } is the Laplace transform of f .
Table of Laplace Transforms:
f (t)                       F (s) = L{f }(s)

1
1                             ,                s>0
s
1
eat                             ,              s>a
s−a
n!
tn , n = 1, 2, 3, ...               ,          s>0
sn+1
b
sin(bt)                              ,         s>0
s 2 + b2
s
cos(bt)                                ,       s>0
s2    + b2
n!
eat tn , n = 1, 2, 3, ...              ,       s>a
(s − a)n+1

b
eat sin(bt)                               ,    s>a
(s − a)2 + b2

s−a
eat cos(bt)                               ,    s>a
(s − a)2 + b2
Note: n! = 1 · 2 · 3...(n − 1)(n). (For example, 1! = 1, 2! = 2, 3! = 6, 4! = 24, 5! = 120, etc.)
Useful Properties:
L{f + g}       L{f } + L{g}
=
L{cf }      cL{f }
=
at
L{e f (t)}(s)     L{f (t)}(s − a)
=
L{f }(s)      sL{f }(s) − f (0)
=
L{f }(s)       s2 L{f }(s) − sf (0) − f (0)
=
L{f (n) }(s)    sn L{f }(s) − sn−1 f (0) − sn−2 f (0) − ... − f (n−1) (0)
=
d
L{tf (t)}(s) = − (L{f }(s))
ds
dn
L{tn f (t)}(s) = (−1)n n (L{f }(s))
ds
The unit step function is written:
0, t < 0
u(t) =
1, 0 ≤ t
(From Theorem 8) For a > 0, we have:
L{f (t − a)u(t − a)}(s) = e−as L{f (t)}(s)

(From Theorem 9) If f has period T and is piecewise continuous on [0, T ], then
T
1
L{f }(s) =                            e−st f (t)dt
1 − e−sT        0

For t > 0, the gamma function Γ(t) is written:
∞
Γ(t) =             e−x xt−1
0

It allows us to deﬁne the Laplace transform of tr , r > −1:
Γ(r + 1)
L{tr }(s) =
sr+1

The convolution of f and g, two piecewise continuous functions deﬁned on [0, ∞) is:
t
(f ∗ g)(t) =           f (t − v)g(v)dv
0

(From Theorems 10 and 11) It satisﬁes the following:
f ∗g    =        g∗f
f ∗ (g + h)    =        (f ∗ g) + (f ∗ h)
(f ∗ g) ∗ h   =        f ∗ (g ∗ h)
f ∗0    =        0
L{f ∗ g}     =        L{f }L{g}

The Dirac delta function δ(t) can be thought of as:
0,                         t=0
δ(t) =
”Something like inﬁnity” , t = 0
It satisﬁes the following:
∞
f (t)δ(t)dt = f (0)
−∞
L{δ(t − a)}(s) = e−as

```
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