φ

and nonzero

p

, substitution

u = pt

with the orientation handled by

∣ p ∣

gives

\int φ (t) δ (pt) d t = \frac{1}{∣ p ∣} φ (0) .

Hence the Dirac delta scaling identity is $δ (pt) = δ (t) /∣ p ∣$ .

ii

Solution

\int_{- \infty}^{\infty} f (t) δ^{'} (t) d t = - \int_{- \infty}^{\infty} f^{'} (t) δ (t) d t = - f^{'} (0) .

For a compactly supported test function

f

⟨ H^{'}, f ⟩ = - ⟨ H, f^{'} ⟩ = - \int_{0}^{\infty} f^{'} (t) d t = f (0) = ⟨ δ, f ⟩ .

Thus this is the distributional derivative of the Heaviside step function: $H^{'} = δ$ .

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