For every integer

j \geq 0

, the

j

t

cos (t cos x)

(cos x)^{j}

1

on the finite interval

[0, π /2]

. This is an integrable dominating function independent of

t

. Repeated differentiation under the integral sign is therefore justified for every order, giving $J_{0} \in C^{\infty} (R)$ .

b

Solution

Differentiating the integral representation gives

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

Now integrate the derivative

\frac{d}{d x} (sin x sin (t cos x)) = cos x sin (t cos x) - t sin^{2} x cos (t cos x) .

The boundary terms vanish at both endpoints. Consequently

J_{0}^{'} (t) = - t (J_{0}^{''} (t) + J_{0} (t))

, proving

t J_{0}^{''} (t) + J_{0}^{'} (t) + t J_{0} (t) = 0.

The identity includes

t = 0

, where it reads

J_{0}^{'} (0) = 0

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