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Integral_Transformations_Laplace_Fourier_Licence_3

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Paris VI - Université Pierre Et Marie Curie

Lecture notes and summary on integral transformations (Laplace and Fourier), all the rules you need to solve your exercises. The words are in basic English but the math part is easy to grasp.

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Uploaded on November 25, 2023
Number of pages 4
Written in 2022/2023
Type Class notes
Professor(s) Patrick
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math
mathematics
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license 3
bac3
university
sorbonne
pierre and marie curie
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Paris VI - Université Pierre et Marie Curie
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Mathématiques
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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

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