Distributional Derivative of Heaviside Step Function times Sine
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Theorem
Let $H$ be the Heaviside step function.
Let $\delta$ be the Dirac delta distribution.
Then in the distributional sense:
- $T'_{\map H x \sin x} = T_{\map H x \cos x} $
Proof
$x \stackrel f {\longrightarrow} \map H x \sin x$ is a continuously differentiable real function on $\R \setminus \set 0$ and possibly has a discontinuity at $x = 0$.
By Differentiable Function as Distribution we have that $T'_f = T_{f'}$.
Moreover:
- $x < 0 \implies \paren {{\map H x} \map \sin x}' = 0$.
- $x > 0 \implies \paren {{\map H x} \map \sin x}' = \cos x$.
Altogether:
- $\forall x \in \R \setminus \set 0 : \paren {{\map H x} \map \sin x}' = \map H x \cos x$
Furthermore:
- $\ds \map f {0^+} - \map f {0^-} = \map H {0^+} \map \sin {0^+} - \map H {0^-} \map \sin {0^-} = 0$
The result follows from the Jump Rule.
$\blacksquare$
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.2$: A glimpse of distribution theory. Derivatives in the distributional sense