Asymptotics for the first zero of the Bessel functions












2












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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 $+infty$?






special-functions asymptotics bessel-functions







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  • $begingroup$
    Have you looked at DLMF 20.21(vii)?
    $endgroup$
    – Somos
    Nov 17 '18 at 20:24


















2












$begingroup$


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 $+infty$?






special-functions asymptotics bessel-functions







share|cite|improve this question









$endgroup$












  • $begingroup$
    Have you looked at DLMF 20.21(vii)?
    $endgroup$
    – Somos
    Nov 17 '18 at 20:24
















2












2








2





$begingroup$


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 $+infty$?






special-functions asymptotics bessel-functions







share|cite|improve this question









$endgroup$




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 $+infty$?






special-functions asymptotics bessel-functions




special-functions asymptotics bessel-functions






share|cite|improve this question













share|cite|improve this question











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asked Nov 17 '18 at 19:43









BazinBazin

8,1671238




8,1671238












  • $begingroup$
    Have you looked at DLMF 20.21(vii)?
    $endgroup$
    – Somos
    Nov 17 '18 at 20:24




















  • $begingroup$
    Have you looked at DLMF 20.21(vii)?
    $endgroup$
    – Somos
    Nov 17 '18 at 20:24


















$begingroup$
Have you looked at DLMF 20.21(vii)?
$endgroup$
– Somos
Nov 17 '18 at 20:24






$begingroup$
Have you looked at DLMF 20.21(vii)?
$endgroup$
– Somos
Nov 17 '18 at 20:24












1 Answer
1






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






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    1 Answer
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    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    5












    $begingroup$

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






    share|cite|improve this answer









    $endgroup$


















      5












      $begingroup$

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






      share|cite|improve this answer









      $endgroup$
















        5












        5








        5





        $begingroup$

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






        share|cite|improve this answer









        $endgroup$



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







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 17 '18 at 19:56









        Francois ZieglerFrancois Ziegler

        19.7k371116




        19.7k371116






























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