Integral of two Bessel functions product times Gaussian
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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 yesterday
Yang
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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 yesterday
Yang
62
New contributor
Yang is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Just a correction... the exponential of e in the integral has a minus sign in front. (corrected in the question...)
– Yang
yesterday
1
In book:Table of Integrals, Series, and Products Eighth Edition6.618 example 5 page 706.
– Mariusz Iwaniuk
yesterday
@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
yesterday
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
yesterday
@PaulEnta.6.633.3it fits more to OP question (First integral).
– Mariusz Iwaniuk
yesterday
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 yesterday
Yang
62
New contributor
Yang is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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 yesterday
Yang
62
New contributor
Yang is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
Yang
62
New contributor
Yang is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited yesterday
asked yesterday
Yang
62
New contributor
Yang is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
Yang
62
asked yesterday
Yang
62
62
New contributor
Yang is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Yang is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Yang is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Just a correction... the exponential of e in the integral has a minus sign in front. (corrected in the question...)
– Yang
yesterday
1
In book:Table of Integrals, Series, and Products Eighth Edition6.618 example 5 page 706.
– Mariusz Iwaniuk
yesterday
@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
yesterday
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
yesterday
@PaulEnta.6.633.3it fits more to OP question (First integral).
– Mariusz Iwaniuk
yesterday
add a comment |
Just a correction... the exponential of e in the integral has a minus sign in front. (corrected in the question...)
– Yang
yesterday
1
In book:Table of Integrals, Series, and Products Eighth Edition6.618 example 5 page 706.
– Mariusz Iwaniuk
yesterday
@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
yesterday
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
yesterday
@PaulEnta.6.633.3it fits more to OP question (First integral).
– Mariusz Iwaniuk
yesterday
Just a correction... the exponential of e in the integral has a minus sign in front. (corrected in the question...)
– Yang
yesterday
Just a correction... the exponential of e in the integral has a minus sign in front. (corrected in the question...)
– Yang
yesterday
1
1
In book:
Table of Integrals, Series, and Products Eighth Edition 6.618 example 5 page 706.– Mariusz Iwaniuk
yesterday
In book:
Table of Integrals, Series, and Products Eighth Edition 6.618 example 5 page 706.– Mariusz Iwaniuk
yesterday
@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
yesterday
@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
yesterday
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
yesterday
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
yesterday
@PaulEnta.
6.633.3 it fits more to OP question (First integral).– Mariusz Iwaniuk
yesterday
@PaulEnta.
6.633.3 it fits more to OP question (First integral).– Mariusz Iwaniuk
yesterday
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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
yesterday
1
In book:
Table of Integrals, Series, and Products Eighth Edition6.618 example 5 page 706.– Mariusz Iwaniuk
yesterday
@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
yesterday
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
yesterday
@PaulEnta.
6.633.3it fits more to OP question (First integral).– Mariusz Iwaniuk
yesterday