# Asymptotics for the first zero of the Bessel functions

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 $$+\infty$$?

• Have you looked at DLMF 20.21(vii)? Nov 17, 2018 at 20:24

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