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
asked Nov 17 at 19:43
Bazin
8,0871236
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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
asked Nov 17 at 19:43
Bazin
8,0871236
Have you looked at DLMF 20.21(vii)?
– Somos
Nov 17 at 20:24
add a comment |
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
asked Nov 17 at 19:43
Bazin
8,0871236
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
asked Nov 17 at 19:43
Bazin
8,0871236
asked Nov 17 at 19:43
Bazin
8,0871236
asked Nov 17 at 19:43
Bazin
8,0871236
asked Nov 17 at 19:43
Bazin
8,0871236
asked Nov 17 at 19:43
Bazin
8,0871236
8,0871236
Have you looked at DLMF 20.21(vii)?
– Somos
Nov 17 at 20:24
add a comment |
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
add a comment |
1 Answer
1
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5
down vote
Watson? (1922, pp. 486, 516, 521) has for the smallest positive zero:
$$
sqrt{nu(nu+2)}<x_nu<sqrt{2(nu+1)(nu+3)},
$$
$$
x_nu= nu+1.855757nu^{1/3}+O(nu^{-1/3}).
$$
answered Nov 17 at 19:56
Francois Ziegler
19.5k371116
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
Watson? (1922, pp. 486, 516, 521) has for the smallest positive zero:
$$
sqrt{nu(nu+2)}<x_nu<sqrt{2(nu+1)(nu+3)},
$$
$$
x_nu= nu+1.855757nu^{1/3}+O(nu^{-1/3}).
$$
answered Nov 17 at 19:56
Francois Ziegler
19.5k371116
add a comment |
up vote
5
down vote
Watson? (1922, pp. 486, 516, 521) has for the smallest positive zero:
$$
sqrt{nu(nu+2)}<x_nu<sqrt{2(nu+1)(nu+3)},
$$
$$
x_nu= nu+1.855757nu^{1/3}+O(nu^{-1/3}).
$$
answered Nov 17 at 19:56
Francois Ziegler
19.5k371116
add a comment |
up vote
5
down vote
up vote
5
down vote
Watson? (1922, pp. 486, 516, 521) has for the smallest positive zero:
$$
sqrt{nu(nu+2)}<x_nu<sqrt{2(nu+1)(nu+3)},
$$
$$
x_nu= nu+1.855757nu^{1/3}+O(nu^{-1/3}).
$$
answered Nov 17 at 19:56
Francois Ziegler
19.5k371116
Watson? (1922, pp. 486, 516, 521) has for the smallest positive zero:
$$
sqrt{nu(nu+2)}<x_nu<sqrt{2(nu+1)(nu+3)},
$$
$$
x_nu= nu+1.855757nu^{1/3}+O(nu^{-1/3}).
$$
answered Nov 17 at 19:56
Francois Ziegler
19.5k371116
answered Nov 17 at 19:56
Francois Ziegler
19.5k371116
answered Nov 17 at 19:56
Francois Ziegler
19.5k371116
answered Nov 17 at 19:56
Francois Ziegler
19.5k371116
19.5k371116
add a comment |
add a comment |
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Have you looked at DLMF 20.21(vii)?
– Somos
Nov 17 at 20:24