 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2010/d/d4/1/a/solution

The

k

(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) .

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