Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 3 / 15A / e

Table of contents
- i e
  - Solution i
- ii e
  - Solution ii

i

Solution

For a test function

f

, substitute

y = n x

and use the Taylor expansion

f (y / n) = f (0) + (y / n) f^{'} (0) + O (n^{- 2})

. Oddness removes the first term, while the Gaussian integral gives

\int y^{2} e^{- y^{2}} d y = π /2

. Hence

\int n^{3} x e^{- n^{2} x^{2}} f (x) d x ⟶ \frac{π}{2} f^{'} (0) = ⟨ - \frac{π}{2} δ^{'}, f ⟩ .

Therefore

n^{3} x e^{- n^{2} x^{2}} ⟶ - \frac{π}{2} δ^{'} (x) .

ii

Solution

Direct integration gives

\frac{1}{π} \int_{0}^{n} cos (k x) d k = \frac{s i n ( n x )}{π x} .

The sinc approximate identity converges distributionally to the Dirac delta function, using the supplied integral of

sin y / y

. Thus the limit is $δ (x)$ .

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