37 lines
534 B
Markdown
37 lines
534 B
Markdown
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# Intégration par parties (IPP)
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Soit $( f, g \in \mathcal{C}^n([a,b]) )$.
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---
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## (IP1)
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$$
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(\mathcal{L}(f))^{(n)}(t) = (-t)^n \, \mathcal{L}(x^n f(x))(t)
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$$
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---
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## (IP2)
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Si \( f \) est de classe \( \mathcal{C}^n \) avec \( f^{(n)} \in \mathcal{L}^1(\mathbb{R}^+) \), alors :
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$$
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\mathcal{L}(f^{(n)})(p)
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= p^n \mathcal{L}(f)(p)
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- p^{n-1} f(0)
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- p^{n-2} f'(0)
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- \dots
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- f^{(n-1)}(0)
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$$
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---
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## (IP3)
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Si \( f \) est la primitive de \( g \) qui s'annule en \( 0 \), alors :
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$$
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\mathcal{L}(f)(p) = \frac{\mathcal{L}(g)(p)}{p}
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$$
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