Bessel function of order 1

alexmahone

Prove that [itex]J_1(x)=\frac{1}{\pi}\int_0^\pi\cos(\theta-x\sin\theta)d\theta[/itex] by showing that the right-hand side satisfies Bessel's equation of order 1 and that the derivative has the value [itex]J_1'(0)[/itex] when [itex]x=0[/itex]. Explain why this constitutes a proof.