Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / D / D4 / 1

Table of contents
- a 1
  - Solution a
- b 1
  - Solution b

a

Solution

The

k

th derivative of the integrand is

(cos x)^{k} cos (t cos x + kπ /2)

, whose absolute value is at most one. The integration interval has finite measure. Repeated application of the differentiation theorem therefore gives

J_{0}^{(k)} (t) = \frac{2}{π} \int_{0}^{π /2} (cos x)^{k} cos (t cos x + kπ /2) d x (k \geq 0) .

Dominated convergence also gives continuity of every derivative, proving infinite differentiability.

b

Solution

From (a), combine the three derivative integrals:

t J_{0}^{''} + J_{0}^{'} + t J_{0} = \frac{2}{π} \int_{0}^{π /2} (t sin^{2} x cos (t cos x) - cos x sin (t cos x)) d x .

The integrand is

- \frac{d}{d x} [sin x sin (t cos x)]

. This bracket vanishes at both endpoints, so the integral is zero. Hence

t J_{0}^{''} + J_{0}^{'} + t J_{0} = 0

, also at

t = 0

by the same formula.

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