Integral_Transformations_Laplace_Fourier_Licence_3

V. Integrals transformations

1. Generalized integrals :

Let be 𝑓: 𝑥 ∈ [𝑎, 𝑏].

𝑏
Definition : ∫𝑎 𝑓(𝑥). 𝑑𝑥 is said to be convergent if its primitive 𝐹(𝑥) has a finite limit as 𝑡 ⟶
𝑏, and it is said to be divergent in the opposite sense. Note that 𝑏 can be finite (a number) or
infinite (∞).
+∞
• Integrals of type ∫𝑎 𝑓(𝑥). 𝑑𝑥 :

❖ Analysis by comparison :

Let 𝑓 and 𝑔 be two functions such that : ∀𝑥 ≤ 𝑎 ; 0 ≤ 𝑓(𝑥) ≤ 𝑔(𝑥)
+∞ +∞
▪ If ∫𝑎 𝑔(𝑥). 𝑑𝑥 converges, then ∫𝑎 𝑓(𝑥). 𝑑𝑥 converges.
+∞ +∞
▪ If ∫𝑎 𝑔(𝑥). 𝑑𝑥 diverges, then ∫𝑎 𝑓(𝑥). 𝑑𝑥 diverges.

❖ Analysis by equivalence :

Let 𝑓 and 𝑔 be two functions that are equivalent when 𝑥 ⟶ +∞. Then
+∞ +∞
∫𝑎 𝑓(𝑥). 𝑑𝑥 and ∫𝑎 𝑔(𝑥). 𝑑𝑥 are of the same nature.

+∞ 𝑑𝑥
❖ ∫𝑎 ; 𝑎 > 0, is convergente if and only if 𝛼 > 1.
𝑥𝛼

2. Laplace transformation :

Principle : The Laplace transform transforms a time-domain function 𝑓(𝑡). 𝑢(𝑡) into a
complex-valued function 𝐹(𝑝) ; 𝑝 ∈ ℂ, such that :
+∞

𝑭(𝒑) = ℒ[𝒇(𝒕). 𝒖(𝒕)] = ∫ 𝒇(𝒕)𝒆−𝒑𝒕 . 𝒅𝒕
𝟎

𝑝 = 𝑥 + 𝑖𝑦 is called the original, and ℒ(𝑝) is its image.

❖ Theorem 1 : Linearity

ℒ[𝑎𝑓1 + 𝑏𝑓2 ](𝑝) = 𝑎ℒ[𝑓1 ](𝑝) + 𝑏ℒ[𝑓2 ](𝑝)

❖ Theorem 2 : change of scale ; ∀ 𝑓 of summability 𝑥0 .

1 𝑝
▪ ∀𝑎 ∈ ℝ+
∗ ; ℒ[𝑓(𝑎𝑡)](𝑝) = 𝑎 ℒ [𝑓 (𝑎)] , if 𝑝 > 𝑎𝑥0 .
▪ ∀𝑎 ∈ ℝ ; ℒ[𝑒 𝑎𝑡 𝑓(𝑡)](𝑝) = ℒ[𝑓(𝑝 − 𝑎)] , ∀𝑝 > 𝑎 + 𝑥0 .

❖ Theorem 3 : derivative of the transform ; ∀ 𝑓
+∞
𝑑𝑛
𝑛
(ℒ(𝑓)(𝑝)) = ∫ (−𝑡 𝑛 )𝑒 −𝑝𝑡 𝑓(𝑡). 𝑑𝑡
𝑑𝑝
0