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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 $+\infty$?

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2 Answers 2

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Watson? (1922, pp. 486, 516, 521) has for the smallest positive zero: $$ \sqrt{\nu(\nu+2)}<x_\nu<\sqrt{2(\nu+1)(\nu+3)}, $$ $$ x_\nu= \nu+1.855757\nu^{1/3}+O(\nu^{-1/3}). $$

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(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.

[2] https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/further-method-for-the-evaluation-of-zeros-of-bessel-functions-and-some-new-asymptotic-expansions-for-zeros-of-functions-of-large-order/3A90541296335E9F85872E91C84098B1

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