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#51 2016-10-10 21:28:56

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Sum of an integral involving Bessel functions

That is very true. So if we can not show analytically that it converges and we can not even show it empirically then how?

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#52 2016-10-10 23:17:31

zetafunc
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Re: Sum of an integral involving Bessel functions

It may be possible analytically with a particular change of co-ordinates. I'm going to talk to my supervisor in a couple of hours to find out how to do it.

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#53 2016-10-11 01:53:08

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Sum of an integral involving Bessel functions

Okay, good luck. Let me know what he says.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#54 2016-10-11 03:40:24

zetafunc
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Re: Sum of an integral involving Bessel functions

He said my method was wrong but somehow I ended up with the right integral, except the limits are R^d rather than [0,2pi)^d.

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#55 2016-10-11 03:46:46

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Sum of an integral involving Bessel functions

R^d

I am sorry, I am not following that. What does that mean for Mathematica?

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#56 2016-10-11 04:03:05

zetafunc
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Re: Sum of an integral involving Bessel functions

So if d = 2 it would be a double integral such as the one in post #1 but without the sum.

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#57 2016-10-11 04:05:52

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Sum of an integral involving Bessel functions

That might be a lot easier to do. Anyway, did he give some hint as to what the value of the integral will be. Smaller than 1?

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#58 2016-10-11 04:09:14

zetafunc
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Re: Sum of an integral involving Bessel functions

He has no idea. I think I did try this integral once but I can't remember what the results were, it might be useful to investigate for smaller dimensions.

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#59 2016-10-11 04:11:01

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Sum of an integral involving Bessel functions

Okay, post your code if you get some.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#60 2016-10-31 11:40:15

zetafunc
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Re: Sum of an integral involving Bessel functions

Managed to show the integral converges in all dimensions
remains unknown. That is, the integral

converges in those dimensions.

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#61 2016-10-31 13:40:57

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Sum of an integral involving Bessel functions

The integral from post #3?

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#62 2016-10-31 19:09:46

zetafunc
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Re: Sum of an integral involving Bessel functions

Yeah, but without the sum in front. (And we integrate over all of R^d instead of [1,infinity)^d.)

I suspect that the integral in the case d = 2 does not converge, though I've yet to prove that concretely.

Last edited by zetafunc (2016-11-01 00:17:13)

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#63 2016-11-01 01:42:57

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Sum of an integral involving Bessel functions

Why do you think that?

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#64 2016-11-01 02:53:23

zetafunc
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Re: Sum of an integral involving Bessel functions

I can't say anything concrete, but I think I might be able to show that the integral is bounded below by something which is divergent.

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#65 2016-11-01 03:16:21

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Sum of an integral involving Bessel functions

That would be a good way to do it.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#66 2016-11-01 07:17:35

zetafunc
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Re: Sum of an integral involving Bessel functions

I've posted a related question on MO. Received 1 upvote so far, but no responses. My supervisor will not be able to help for the next fortnight or so because he is going to the US.

The question can be found here.

Here is a simplified version of the same question on MathSE: http://math.stackexchange.com/questions … -variables

Last edited by zetafunc (2016-11-01 09:33:48)

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#67 2016-11-01 10:21:08

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Sum of an integral involving Bessel functions

I upvoted it too but I do not see any responses yet.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#68 2016-11-02 03:33:32

zetafunc
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Re: Sum of an integral involving Bessel functions

Thanks. The lack of responses on MO is not a good sign -- it means that the only one who I can get help from is my supervisor.

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#69 2016-11-02 05:08:32

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Sum of an integral involving Bessel functions

Are you sure he can answer the question?

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#70 2016-11-02 06:33:50

zetafunc
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Re: Sum of an integral involving Bessel functions

Yes, because he tried to show me a few weeks ago but he didn't go into detail. Essentially I don't understand how to take the Cauchy product of series whose indices vary over any arbitrary rational lattice. A simplified version of what I need to know is asked here, but I also have received no responses: http://math.stackexchange.com/questions … -variables

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#71 2016-11-02 09:36:16

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Sum of an integral involving Bessel functions

But is that double integral correct?

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#72 2016-11-02 09:43:16

zetafunc
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Re: Sum of an integral involving Bessel functions

Which double integral?

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#73 2016-11-02 09:52:15

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Sum of an integral involving Bessel functions

The one in the other thread?! Excuse me, but I thought both threads were essentially talking about the same or almost the same one?!

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#74 2016-11-02 09:54:03

zetafunc
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Re: Sum of an integral involving Bessel functions

The integrands are the same, but the integral in this thread ranges over [0,2pi)^2, whilst the other one ranges over all of R^2.

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#75 2016-11-02 09:59:34

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Sum of an integral involving Bessel functions

I remember the code I was looking at:

``````Sum[Abs[Integrate[((x^2 + y^2)^(-1/2))*(((b - x)^2 + (c - y)^2)^(-1/2))*
BesselJ[1, k*Sqrt[x^2 + y^2]]*
BesselJ[1, k*Sqrt[(b - x)^2 + (y - c)^2]], {x, 0, 2*Pi}, {y, 0, 2*Pi}]^2], {b, 1, Infinity}, {c, 1, Infinity}]``````

I remember that it did not seem to be converging. I asked if your supervisor knew what it converged to. I do not remember your answer...

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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