The roots of Bessel functions
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In Abramowitz Stegun 1972 there is an inequality (9.5.2) for roots of Bessel functions and their derivatives (n is positive): $$ n leq j^{'}_{n,1} <y^{}_{n,1} < y^{'}_{n,1} < j^{}_{n,1} < j^{'}_{n,2} <... $$
Here first index denotes the order of Bessel function, second - the order of root and " ' " denotes that this is a root of derivative of Bessel function. Can anybody help me and tell how this inequality can be proven, or where can i read about this? Also, I'm very interested even in left part of inequality, when n is integer:
$$ n leq j^{'}_{n,1} $$
differential-equations special-functions
asked Nov 18 at 21:30
user513532
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In Abramowitz Stegun 1972 there is an inequality (9.5.2) for roots of Bessel functions and their derivatives (n is positive): $$ n leq j^{'}_{n,1} <y^{}_{n,1} < y^{'}_{n,1} < j^{}_{n,1} < j^{'}_{n,2} <... $$
Here first index denotes the order of Bessel function, second - the order of root and " ' " denotes that this is a root of derivative of Bessel function. Can anybody help me and tell how this inequality can be proven, or where can i read about this? Also, I'm very interested even in left part of inequality, when n is integer:
$$ n leq j^{'}_{n,1} $$
differential-equations special-functions
asked Nov 18 at 21:30
user513532
11
I know next to nothing about this topic, but the place I'd begin is Chapter XV: The Zeros of Bessel Functions on pp. 477-521 of A Treatise on the Theory of Bessel Functions by George N. Watson. FYI, I know about this reference for your question because of some investigations I made a few years ago regarding the positive solutions of $tan x = x$, which are also the positive solutions of $J_{frac{3}{2}}(x) = 0.$
– Dave L. Renfro
Nov 18 at 22:02
Very grateful, in Watson's book this inequality is proven!
– user513532
Nov 18 at 22:56
Could you post an answer to your question including the reference to Watson's book (just copy parts of the first sentence in my comment) and where in that reference the answer to your question is given? Also, if it doesn't take more than a few minutes, maybe give an indication of how the result is proved, and why someone would want to know something like this? This way your question won't be answer-less, and my simply posting my comment as an answer would not be particuarly useful to anyone later searching for information related to this topic in Stack Exchange.
– Dave L. Renfro
Nov 19 at 8:56
Perhaps better would be to put the part about why someone would be interested in this result (what Watson says, if anything, and/or you) in your question. Right now your question comes close to meeting the typical standards in which the questioner is expected to indicate partial work towards solving the problem (probably not needed here, as this doesn't appear to be homework) and to indicate how the problem arises (sometimes they're naturally occurring, but this one doesn't seem so).
– Dave L. Renfro
Nov 19 at 9:01
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
In Abramowitz Stegun 1972 there is an inequality (9.5.2) for roots of Bessel functions and their derivatives (n is positive): $$ n leq j^{'}_{n,1} <y^{}_{n,1} < y^{'}_{n,1} < j^{}_{n,1} < j^{'}_{n,2} <... $$
Here first index denotes the order of Bessel function, second - the order of root and " ' " denotes that this is a root of derivative of Bessel function. Can anybody help me and tell how this inequality can be proven, or where can i read about this? Also, I'm very interested even in left part of inequality, when n is integer:
$$ n leq j^{'}_{n,1} $$
differential-equations special-functions
asked Nov 18 at 21:30
user513532
11
In Abramowitz Stegun 1972 there is an inequality (9.5.2) for roots of Bessel functions and their derivatives (n is positive): $$ n leq j^{'}_{n,1} <y^{}_{n,1} < y^{'}_{n,1} < j^{}_{n,1} < j^{'}_{n,2} <... $$
Here first index denotes the order of Bessel function, second - the order of root and " ' " denotes that this is a root of derivative of Bessel function. Can anybody help me and tell how this inequality can be proven, or where can i read about this? Also, I'm very interested even in left part of inequality, when n is integer:
$$ n leq j^{'}_{n,1} $$
differential-equations special-functions
differential-equations special-functions
asked Nov 18 at 21:30
user513532
11
asked Nov 18 at 21:30
user513532
11
asked Nov 18 at 21:30
user513532
11
asked Nov 18 at 21:30
user513532
11
asked Nov 18 at 21:30
user513532
11
11
I know next to nothing about this topic, but the place I'd begin is Chapter XV: The Zeros of Bessel Functions on pp. 477-521 of A Treatise on the Theory of Bessel Functions by George N. Watson. FYI, I know about this reference for your question because of some investigations I made a few years ago regarding the positive solutions of $tan x = x$, which are also the positive solutions of $J_{frac{3}{2}}(x) = 0.$
– Dave L. Renfro
Nov 18 at 22:02
Very grateful, in Watson's book this inequality is proven!
– user513532
Nov 18 at 22:56
Could you post an answer to your question including the reference to Watson's book (just copy parts of the first sentence in my comment) and where in that reference the answer to your question is given? Also, if it doesn't take more than a few minutes, maybe give an indication of how the result is proved, and why someone would want to know something like this? This way your question won't be answer-less, and my simply posting my comment as an answer would not be particuarly useful to anyone later searching for information related to this topic in Stack Exchange.
– Dave L. Renfro
Nov 19 at 8:56
Perhaps better would be to put the part about why someone would be interested in this result (what Watson says, if anything, and/or you) in your question. Right now your question comes close to meeting the typical standards in which the questioner is expected to indicate partial work towards solving the problem (probably not needed here, as this doesn't appear to be homework) and to indicate how the problem arises (sometimes they're naturally occurring, but this one doesn't seem so).
– Dave L. Renfro
Nov 19 at 9:01
add a comment |
I know next to nothing about this topic, but the place I'd begin is Chapter XV: The Zeros of Bessel Functions on pp. 477-521 of A Treatise on the Theory of Bessel Functions by George N. Watson. FYI, I know about this reference for your question because of some investigations I made a few years ago regarding the positive solutions of $tan x = x$, which are also the positive solutions of $J_{frac{3}{2}}(x) = 0.$
– Dave L. Renfro
Nov 18 at 22:02
Very grateful, in Watson's book this inequality is proven!
– user513532
Nov 18 at 22:56
Could you post an answer to your question including the reference to Watson's book (just copy parts of the first sentence in my comment) and where in that reference the answer to your question is given? Also, if it doesn't take more than a few minutes, maybe give an indication of how the result is proved, and why someone would want to know something like this? This way your question won't be answer-less, and my simply posting my comment as an answer would not be particuarly useful to anyone later searching for information related to this topic in Stack Exchange.
– Dave L. Renfro
Nov 19 at 8:56
Perhaps better would be to put the part about why someone would be interested in this result (what Watson says, if anything, and/or you) in your question. Right now your question comes close to meeting the typical standards in which the questioner is expected to indicate partial work towards solving the problem (probably not needed here, as this doesn't appear to be homework) and to indicate how the problem arises (sometimes they're naturally occurring, but this one doesn't seem so).
– Dave L. Renfro
Nov 19 at 9:01
I know next to nothing about this topic, but the place I'd begin is Chapter XV: The Zeros of Bessel Functions on pp. 477-521 of A Treatise on the Theory of Bessel Functions by George N. Watson. FYI, I know about this reference for your question because of some investigations I made a few years ago regarding the positive solutions of $tan x = x$, which are also the positive solutions of $J_{frac{3}{2}}(x) = 0.$
– Dave L. Renfro
Nov 18 at 22:02
I know next to nothing about this topic, but the place I'd begin is Chapter XV: The Zeros of Bessel Functions on pp. 477-521 of A Treatise on the Theory of Bessel Functions by George N. Watson. FYI, I know about this reference for your question because of some investigations I made a few years ago regarding the positive solutions of $tan x = x$, which are also the positive solutions of $J_{frac{3}{2}}(x) = 0.$
– Dave L. Renfro
Nov 18 at 22:02
Very grateful, in Watson's book this inequality is proven!
– user513532
Nov 18 at 22:56
Very grateful, in Watson's book this inequality is proven!
– user513532
Nov 18 at 22:56
Could you post an answer to your question including the reference to Watson's book (just copy parts of the first sentence in my comment) and where in that reference the answer to your question is given? Also, if it doesn't take more than a few minutes, maybe give an indication of how the result is proved, and why someone would want to know something like this? This way your question won't be answer-less, and my simply posting my comment as an answer would not be particuarly useful to anyone later searching for information related to this topic in Stack Exchange.
– Dave L. Renfro
Nov 19 at 8:56
Could you post an answer to your question including the reference to Watson's book (just copy parts of the first sentence in my comment) and where in that reference the answer to your question is given? Also, if it doesn't take more than a few minutes, maybe give an indication of how the result is proved, and why someone would want to know something like this? This way your question won't be answer-less, and my simply posting my comment as an answer would not be particuarly useful to anyone later searching for information related to this topic in Stack Exchange.
– Dave L. Renfro
Nov 19 at 8:56
Perhaps better would be to put the part about why someone would be interested in this result (what Watson says, if anything, and/or you) in your question. Right now your question comes close to meeting the typical standards in which the questioner is expected to indicate partial work towards solving the problem (probably not needed here, as this doesn't appear to be homework) and to indicate how the problem arises (sometimes they're naturally occurring, but this one doesn't seem so).
– Dave L. Renfro
Nov 19 at 9:01
Perhaps better would be to put the part about why someone would be interested in this result (what Watson says, if anything, and/or you) in your question. Right now your question comes close to meeting the typical standards in which the questioner is expected to indicate partial work towards solving the problem (probably not needed here, as this doesn't appear to be homework) and to indicate how the problem arises (sometimes they're naturally occurring, but this one doesn't seem so).
– Dave L. Renfro
Nov 19 at 9:01
add a comment |
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I know next to nothing about this topic, but the place I'd begin is Chapter XV: The Zeros of Bessel Functions on pp. 477-521 of A Treatise on the Theory of Bessel Functions by George N. Watson. FYI, I know about this reference for your question because of some investigations I made a few years ago regarding the positive solutions of $tan x = x$, which are also the positive solutions of $J_{frac{3}{2}}(x) = 0.$
– Dave L. Renfro
Nov 18 at 22:02
Very grateful, in Watson's book this inequality is proven!
– user513532
Nov 18 at 22:56
Could you post an answer to your question including the reference to Watson's book (just copy parts of the first sentence in my comment) and where in that reference the answer to your question is given? Also, if it doesn't take more than a few minutes, maybe give an indication of how the result is proved, and why someone would want to know something like this? This way your question won't be answer-less, and my simply posting my comment as an answer would not be particuarly useful to anyone later searching for information related to this topic in Stack Exchange.
– Dave L. Renfro
Nov 19 at 8:56
Perhaps better would be to put the part about why someone would be interested in this result (what Watson says, if anything, and/or you) in your question. Right now your question comes close to meeting the typical standards in which the questioner is expected to indicate partial work towards solving the problem (probably not needed here, as this doesn't appear to be homework) and to indicate how the problem arises (sometimes they're naturally occurring, but this one doesn't seem so).
– Dave L. Renfro
Nov 19 at 9:01