Integral of two Bessel functions product times Gaussian
up vote
1
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Does anyone have a clue about how to solve this integral? Will it have a closed form?
$int_0^infty e^{-x^2}J_n(ax)J_n(bx)dx$
I've been searching materials and papers for a while, and did find anything about it... People had this integral
$int_0^infty x^2e^{-x^2}J_n(ax)J_n(bx)dx$
solved instead of the above form.
integration bessel-functions gaussian-integral
asked Nov 16 at 7:08
Yang
42
add a comment |
up vote
1
down vote
favorite
Does anyone have a clue about how to solve this integral? Will it have a closed form?
$int_0^infty e^{-x^2}J_n(ax)J_n(bx)dx$
I've been searching materials and papers for a while, and did find anything about it... People had this integral
$int_0^infty x^2e^{-x^2}J_n(ax)J_n(bx)dx$
solved instead of the above form.
integration bessel-functions gaussian-integral
asked Nov 16 at 7:08
Yang
42
Just a correction... the exponential of e in the integral has a minus sign in front. (corrected in the question...)
– Yang
Nov 16 at 7:13
1
In book:Table of Integrals, Series, and Products Eighth Edition6.618 example 5 page 706.
– Mariusz Iwaniuk
Nov 16 at 7:23
@MariuszIwaniuk wouldn't it be (6.633.1) rather than (6.618.5) where the Bessel functions have identical variables $J_mu(beta x)J_nu(beta x)$?
– Paul Enta
Nov 16 at 16:28
I can only find the PDF of the seventh edition. Then, I think the equations you are talking about is 6.633 example 1, page 707 in the seventh edition. Thanks a lot for your help! (But sadly, the result has an infinite summation inside, which is still very hard to handle. :( )
– Yang
Nov 16 at 17:26
@PaulEnta.6.633.3it fits more to OP question (First integral).
– Mariusz Iwaniuk
Nov 16 at 17:38
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Does anyone have a clue about how to solve this integral? Will it have a closed form?
$int_0^infty e^{-x^2}J_n(ax)J_n(bx)dx$
I've been searching materials and papers for a while, and did find anything about it... People had this integral
$int_0^infty x^2e^{-x^2}J_n(ax)J_n(bx)dx$
solved instead of the above form.
integration bessel-functions gaussian-integral
asked Nov 16 at 7:08
Yang
42
Does anyone have a clue about how to solve this integral? Will it have a closed form?
$int_0^infty e^{-x^2}J_n(ax)J_n(bx)dx$
I've been searching materials and papers for a while, and did find anything about it... People had this integral
$int_0^infty x^2e^{-x^2}J_n(ax)J_n(bx)dx$
solved instead of the above form.
integration bessel-functions gaussian-integral
integration bessel-functions gaussian-integral
asked Nov 16 at 7:08
Yang
42
asked Nov 16 at 7:08
Yang
42
edited Nov 16 at 7:14
asked Nov 16 at 7:08
Yang
42
asked Nov 16 at 7:08
Yang
42
asked Nov 16 at 7:08
Yang
42
42
Just a correction... the exponential of e in the integral has a minus sign in front. (corrected in the question...)
– Yang
Nov 16 at 7:13
1
In book:Table of Integrals, Series, and Products Eighth Edition6.618 example 5 page 706.
– Mariusz Iwaniuk
Nov 16 at 7:23
@MariuszIwaniuk wouldn't it be (6.633.1) rather than (6.618.5) where the Bessel functions have identical variables $J_mu(beta x)J_nu(beta x)$?
– Paul Enta
Nov 16 at 16:28
I can only find the PDF of the seventh edition. Then, I think the equations you are talking about is 6.633 example 1, page 707 in the seventh edition. Thanks a lot for your help! (But sadly, the result has an infinite summation inside, which is still very hard to handle. :( )
– Yang
Nov 16 at 17:26
@PaulEnta.6.633.3it fits more to OP question (First integral).
– Mariusz Iwaniuk
Nov 16 at 17:38
add a comment |
Just a correction... the exponential of e in the integral has a minus sign in front. (corrected in the question...)
– Yang
Nov 16 at 7:13
1
In book:Table of Integrals, Series, and Products Eighth Edition6.618 example 5 page 706.
– Mariusz Iwaniuk
Nov 16 at 7:23
@MariuszIwaniuk wouldn't it be (6.633.1) rather than (6.618.5) where the Bessel functions have identical variables $J_mu(beta x)J_nu(beta x)$?
– Paul Enta
Nov 16 at 16:28
I can only find the PDF of the seventh edition. Then, I think the equations you are talking about is 6.633 example 1, page 707 in the seventh edition. Thanks a lot for your help! (But sadly, the result has an infinite summation inside, which is still very hard to handle. :( )
– Yang
Nov 16 at 17:26
@PaulEnta.6.633.3it fits more to OP question (First integral).
– Mariusz Iwaniuk
Nov 16 at 17:38
Just a correction... the exponential of e in the integral has a minus sign in front. (corrected in the question...)
– Yang
Nov 16 at 7:13
Just a correction... the exponential of e in the integral has a minus sign in front. (corrected in the question...)
– Yang
Nov 16 at 7:13
1
1
In book:
Table of Integrals, Series, and Products Eighth Edition 6.618 example 5 page 706.– Mariusz Iwaniuk
Nov 16 at 7:23
In book:
Table of Integrals, Series, and Products Eighth Edition 6.618 example 5 page 706.– Mariusz Iwaniuk
Nov 16 at 7:23
@MariuszIwaniuk wouldn't it be (6.633.1) rather than (6.618.5) where the Bessel functions have identical variables $J_mu(beta x)J_nu(beta x)$?
– Paul Enta
Nov 16 at 16:28
@MariuszIwaniuk wouldn't it be (6.633.1) rather than (6.618.5) where the Bessel functions have identical variables $J_mu(beta x)J_nu(beta x)$?
– Paul Enta
Nov 16 at 16:28
I can only find the PDF of the seventh edition. Then, I think the equations you are talking about is 6.633 example 1, page 707 in the seventh edition. Thanks a lot for your help! (But sadly, the result has an infinite summation inside, which is still very hard to handle. :( )
– Yang
Nov 16 at 17:26
I can only find the PDF of the seventh edition. Then, I think the equations you are talking about is 6.633 example 1, page 707 in the seventh edition. Thanks a lot for your help! (But sadly, the result has an infinite summation inside, which is still very hard to handle. :( )
– Yang
Nov 16 at 17:26
@PaulEnta.
6.633.3 it fits more to OP question (First integral).– Mariusz Iwaniuk
Nov 16 at 17:38
@PaulEnta.
6.633.3 it fits more to OP question (First integral).– Mariusz Iwaniuk
Nov 16 at 17:38
add a comment |
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Just a correction... the exponential of e in the integral has a minus sign in front. (corrected in the question...)
– Yang
Nov 16 at 7:13
1
In book:
Table of Integrals, Series, and Products Eighth Edition6.618 example 5 page 706.– Mariusz Iwaniuk
Nov 16 at 7:23
@MariuszIwaniuk wouldn't it be (6.633.1) rather than (6.618.5) where the Bessel functions have identical variables $J_mu(beta x)J_nu(beta x)$?
– Paul Enta
Nov 16 at 16:28
I can only find the PDF of the seventh edition. Then, I think the equations you are talking about is 6.633 example 1, page 707 in the seventh edition. Thanks a lot for your help! (But sadly, the result has an infinite summation inside, which is still very hard to handle. :( )
– Yang
Nov 16 at 17:26
@PaulEnta.
6.633.3it fits more to OP question (First integral).– Mariusz Iwaniuk
Nov 16 at 17:38