Category:Composition of Dirac Delta Distribution with Function with Simple Zero
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This category contains pages concerning Composition of Dirac Delta Distribution with Function with Simple Zero:
Let $\delta \in \map {\DD'} \R$ be the Dirac delta distribution.
Let $\sequence {\map {\delta_n} x}_{n \mathop \in \N}$ be a delta sequence.
Let $f : \R \to \R$ be a real function with a simple zero at $x_0$.
Let $f$ be strictly monotone.
Let $\phi \in \map \DD \R$ be a test function.
Then in the distributional sense it holds that:
- $\ds \map \delta {\map f x} = \frac {\map \delta {x - x_0}}{\size {\map {f'} {x_0}} }$
which can be interpreted as:
- $\ds \int_{-\infty}^\infty \map \delta {\map f x} \map \phi x \rd x = \frac {\map \phi {x_0}}{\size {\map {f'} {x_0}} }$
which more strictly means that:
- $\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map {\delta_n} {\map f x} \map \phi x \rd x = \frac {\map \phi {x_0}}{\size {\map {f'} {x_0}} }$
Pages in category "Composition of Dirac Delta Distribution with Function with Simple Zero"
The following 4 pages are in this category, out of 4 total.
C
- Composition of Dirac Delta Distribution with Function with Simple Zero
- Composition of Dirac Delta Distribution with Function with Simple Zero/Corollary
- Composition of Dirac Delta Distribution with Function with Simple Zero/Proof 1
- Composition of Dirac Delta Distribution with Function with Simple Zero/Proof 2