Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 4 / 5D (Methods)

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- Solution 5D

Solution

Introduce the Cauchy approximate identity

δ_{ε} (x) = \frac{ε}{π ( ε ^{2} + x ^{2} )} .

It is an approximate identity converging to the Dirac delta function

δ

, and

g_{ε} = δ_{ε}^{'}

. For a smooth rapidly decaying test function

ϕ

, integration by parts gives

\int_{- \infty}^{\infty} ϕ (x) g_{ε} (x) d x = - \int_{- \infty}^{\infty} ϕ^{'} (x) δ_{ε} (x) d x ⟶ - ϕ^{'} (0) .

By the definition of the derivative of the Dirac delta,

⟨ δ^{'}, ϕ ⟩ = - ϕ^{'} (0)

. Therefore the limit as a distribution is

g_{ε} ⟶ δ^{'} .

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