Asymptotics for the first zero of the Bessel functions











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

















up vote
2
down vote

favorite












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






















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















up vote
2
down vote

favorite









up vote
2
down vote

favorite











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













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













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asked Nov 17 at 19:43









Bazin

8,0721236




8,0721236












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




















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


















Have you looked at DLMF 20.21(vii)?
– Somos
Nov 17 at 20:24






Have you looked at DLMF 20.21(vii)?
– Somos
Nov 17 at 20:24












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






    share|cite|improve this answer

























      up vote
      5
      down vote













      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























        up vote
        5
        down vote










        up vote
        5
        down vote









        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












        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 at 19:56









        Francois Ziegler

        19.4k370116




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