I'm reading a book on ordinary differential and I'm stuck on one of the exercises.
- (a) Verify that $$J_0(x)=\frac{2}{\pi} \int_0^\frac{\pi}{2} \cos(x\sin(t)) \, dt$$ (b) Deduce from the formula of part (a) that $|J_0(x)| \le 1$ for $x\ge0$
The integral representation in the question is a special case of a more general integral representation for the Bessel function and I know there are several proofs of that already, however I would like to know a more direct proof for the integral in the question. I really don't any idea on where to start, but I guess one could try to show that the integral is a solution to $$x^2y''+xy'+x^2y=0$$ The second part I'm sure is easy to solve using some integration rules, but I'm quite a beginner so I don't what to do.
Thanks for any help.
The second part b) is trivial. For the first part you have $$x^2y'' + xy' + x^2y \\ =-x^2 \int_0^{\pi/2} \cos(x\sin t) \sin^2 t \, {\rm d}t - x \int_0^{\pi/2} \sin(x\sin t) \sin t \, {\rm d}t + x^2 \int_0^{\pi/2} \cos(x\sin t) \, {\rm d}t \\ =x^2 \int_0^{\pi/2} \cos(x\sin t) \cos^2 t \, {\rm d}t + x\left\{ \sin(x\sin t) \cos t \Big|_0^{\pi/2} - x\int_0^{\pi/2} \cos(x\sin t) \cos^2 t \, {\rm d}t \right\} =0$$