Let $J_\nu$ be the standard Bessel function of the first kind and let $x_\nu$ be its smallest zero. Is there a simple reference or result for the asymptotic expansion of $x_\nu$ when $\nu$ goes to $+\infty$?

1$\begingroup$ Have you looked at DLMF 20.21(vii)? $\endgroup$– SomosNov 17, 2018 at 20:24
2 Answers
Watson? (1922, pp. 486, 516, 521) has for the smallest positive zero: $$ \sqrt{\nu(\nu+2)}<x_\nu<\sqrt{2(\nu+1)(\nu+3)}, $$ $$ x_\nu= \nu+1.855757\nu^{1/3}+O(\nu^{1/3}). $$
(I'd post this as a comment on Francois Ziegler's answer but don't have the rep...)
Elbert (https://www.sciencedirect.com/science/article/pii/S037704270000635X#BIB34) cites Tricomi[1] as improving Watson's result to
$$x_v = v + 1.855757 v^{1/3} + 1.03315 v^{1/3} + O(v^{1}) $$
and Olver as improving it further to:
$$x_v = v + 1.855757081 v^{1/3} + 1.0331502 v^{1/3}  0.00397406 v^{1}  0.0907627 v^{5/3} + 0.0433385 v^{7/3} + O(v^{3})$$
(I haven't read Olver's paper too carefully but presumably one can 'turn the crank' on Olver's (8.7) to get further terms as needed)
[1] Tricomi, Francesco Giacomo. Sulle funzioni di Bessel di ordine e argomento pressoché uguali. Tipogr. Bona, 1948.