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Laplace Transform
Definition of Laplace transform of 𝑓(𝑡):
∞

𝐹(𝑠) = 𝐿[𝑓(𝑡)] ≔ ∫ 𝑓(𝑡)𝑒 −𝑠𝑡 𝑑𝑡
0

Definition of the inverse Laplace transform of 𝐹(𝑠):
𝑓(𝑡) = 𝐿−1 [𝐹(𝑠)] =

1 𝜎+𝑗∞
∫
𝐹(𝑠)𝑒 𝑠𝑡 𝑑𝑠
2𝜋𝑗 𝜎−𝑗∞

Note: 𝑠 = 𝜎 + 𝑗𝜔, 𝜎 ≥ 0
---------------------------------------------------------------------------------------------------------------------Examples:
1 𝑡≥0
1
...
) Exponential Function: 𝑓(𝑡) = {𝑒
0
𝑡<0
∞

∞

𝐹(𝑠) = ∫ 𝑒 −𝑎𝑡 𝑒 −𝑠𝑡 𝑑𝑡 = ∫ 𝑒 −(𝑎+𝑠)𝑡 𝑑𝑡 = −
0

0

𝑒 −(𝑎+𝑠)𝑡 ∞
1
1
|𝑡=0 = 0 − (−
)=
𝑠+𝑎
𝑠+𝑎
𝑠+𝑎

3
...
) Ramp Function: 𝑓(𝑡) = {
0 𝑡<0
∞

𝐹(𝑠) = ∫ 𝑡𝑒 −𝑠𝑡 𝑑𝑡 = −
0

∞
𝑡𝑒 −𝑠𝑡 ∞
𝑒 −𝑠𝑡
1
1
|𝑡=0 − ∫ −
𝑑𝑡 = (0 − 0) − 2 𝑒 −𝑠𝑡 |∞
𝑡=0 = 2
𝑠
𝑠
𝑠
𝑠
0

Note: Use Integration by parts
∞
𝑡=0
5
...
) Linearity
𝐿[𝑓1 (𝑡) + 𝑓2 (𝑡)] = 𝐿[𝑓1 (𝑡)] + 𝐿[𝑓2 (𝑡)] = 𝐹1 (𝑠) + 𝐹2 (𝑠)
𝐿[𝑎𝑓1 (𝑡)] = 𝑎𝐿[𝑓1 (𝑡)] = 𝑎𝐹1 (𝑠)
2
...
) Exponential Scaling
Let 𝑔(𝑡) = 𝑒 𝑎𝑡 𝑓(𝑡) for some scalar 𝑎
𝐺(𝑠) = 𝐹(𝑠 − 𝑎)
4
...
) Derivative
Let 𝑔(𝑡) =

𝑑
𝑓(𝑡)
𝑑𝑡

Let 𝑔(𝑡) =

𝑑2
𝑓(𝑡)
𝑑𝑡 2

𝐿[𝑔(𝑡)] = 𝑠𝐹(𝑠) − 𝑓(0)
𝐿[𝑔(𝑡)] = 𝑠 2 𝐹(𝑠) − 𝑠𝑓(0) − 𝑓′(0)

𝑑3

Let 𝑔(𝑡) = 𝑑𝑡 3 𝑓(𝑡)

𝐿[𝑔(𝑡)] = 𝑠 3 𝐹(𝑠) − 𝑠 2 𝑓(0) − 𝑠𝑓 ′ (0) − 𝑓′′(0)

6
...
) Multiplication by t
Let 𝑔(𝑡) = 𝑡𝑓(𝑡)
𝐿[𝑔(𝑡)] = −

𝑑
𝐹(𝑠)
𝑑𝑠

8

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