On the proof that the Bessel functions of the first kind tend to zero as $x$ tends to infinity

Question

I am looking for a proof that $\lim_{x \rightarrow \infty} J_\nu (x) = 0$ for $\nu \geqslant 0.$ I can see this must be true from the asymptotic expansion for $J_\nu (x)$, namely $$J_\nu (x) \sim \sqrt{\frac{2}{\pi x}} \cos \left (x - \frac{\pi}{4} - \frac{\nu \pi}{2} \right ),$$ but wish to know if it is possible to prove this result more directly using say either the series representation or one of the integral representations for $J_\nu (x)$ without having to have to prove the asymptotic result first?

Answer 1

L. J. Landau published in J. London Math. Soc. (2) 61 (2000) 197–215 this result: |J_p(x)| <= c/|x|^(1/3) for all real x and p > 0 where c = 0.785 746 870 4… is the best possible constant. This reference settles the issue of limit zero at infinity for p > 0. Most find the proof daunting or completely inaccessible.

A simple approach for integers p: expand J_p(x) in terms of J_0(x) and J_1(x) using the basic identity J_{p+1} = (2p/x)J_p - J_{p-1}$. For instance,

J_4(x) = (48/x^3 - 8/x) J_1(x) + (1 - 24/x^2) J_0(x)

The limit problem for integer p > 0 is then reduced to proving that J_1(x) and J_0(x) have limit zero at x=infinity.

Computer plots of J_1 and J_0 should be convincing. A valid proof of limit zero at infinity for J_0 and J_1 using the series alone seems unlikely, in view of Landau's work. Bessel's integral representation of J_p (Wikipedia) might be productive, but for me not: