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#1 2016-10-09 01:26:13

zetafunc
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Registered: 2014-05-21
Posts: 1,831
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Sum of an integral involving Bessel functions

I am interested in the following sum:

where
is a constant, and
denotes the Bessel function of the first kind. This is a special case of a more general sum I'd like to consider, with
:

where
denotes the standard Euclidean norm on
, i.e.
. I've tried computing this in a few different ways using Mathematica. The first way to get rid of the Bessel functions is to use the bound
for some constant
depending on
. However, this may be dangerous, since by taking absolute values of the Bessel functions, we lose the ability to take advantage of any positive-negative cancellation that occurs. Mathematica doesn't seem to be able to compute the integral for
, although one can get a numerical result by replacing
with numbers instead (but since I want to sum over
, this is not entirely helpful). We can also use the asymptotic expansions of
, and in the case of
, the Bessel function
has a very simple closed form:

and in general, there are also finite sum expansions for half-integer values of Bessel functions. But I haven't managed to get any semblance of a result, using any of these methods.

Can anyone find a way of computing this sum for
, or perhaps a better method for general
?

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#2 2016-10-09 03:06:33

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 108,752

Re: Sum of an integral involving Bessel functions

Hi;

Can you please show me your M code?


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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#3 2016-10-09 03:14:50

zetafunc
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Registered: 2014-05-21
Posts: 1,831
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Re: Sum of an integral involving Bessel functions

The input for that sum for d = 2 (restricted to positive integers) is

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}]

but that will not give an answer, I don't think. I'm currently trying to use NSum/NIntegrate to try to find some partial sums given parameters
.

Last edited by zetafunc (2016-10-09 03:24:16)

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#4 2016-10-09 03:22:39

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 108,752

Re: Sum of an integral involving Bessel functions

I am getting a syntax error out of that. Please check your brackets.


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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#5 2016-10-09 03:24:40

zetafunc
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Registered: 2014-05-21
Posts: 1,831
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Re: Sum of an integral involving Bessel functions

Sorry, I fixed it. There was a ] missing.

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#6 2016-10-09 03:39:09

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 108,752

Re: Sum of an integral involving Bessel functions

That is a very tough problem and may not have a closed form. What kind of answer are you looking for?


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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#7 2016-10-09 03:42:35

zetafunc
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Registered: 2014-05-21
Posts: 1,831
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Re: Sum of an integral involving Bessel functions

I do not necessarily need to know the exact sum. I do know that it should converge, but I am really trying to find a bound for it in terms of
.

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#8 2016-10-09 03:46:33

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 108,752

Re: Sum of an integral involving Bessel functions

Perhaps I can get something out of this. First, I would like to test empirically your assertion that it converges.


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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#9 2016-10-09 03:50:15

zetafunc
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Registered: 2014-05-21
Posts: 1,831
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Re: Sum of an integral involving Bessel functions

It is possible it may diverge, but if that is the case, it means that I have done something wrong (it should be true that the above sum is actually
).

Last edited by zetafunc (2016-10-09 03:50:57)

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#10 2016-10-09 03:51:22

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 108,752

Re: Sum of an integral involving Bessel functions

One problem is that I do not know what k is. Can you say something about k?


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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#11 2016-10-09 03:53:53

zetafunc
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Registered: 2014-05-21
Posts: 1,831
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Re: Sum of an integral involving Bessel functions

Sorry, the k should be a
. I just entered k because it was easier than entering
into M.

Last edited by zetafunc (2016-10-09 03:54:07)

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#12 2016-10-09 03:56:12

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 108,752

Re: Sum of an integral involving Bessel functions

So, you want k to be another free variable or I am hoping we can at least bound 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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#13 2016-10-09 04:07:13

zetafunc
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Registered: 2014-05-21
Posts: 1,831
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Re: Sum of an integral involving Bessel functions

We can use the bound mentioned in post #1 but it may cause the integral to either converge or diverge. If d = 2 then, after bounding the Bessel functions, one gets:

(I left out the
as neither the sum nor integral depend on it now, if we choose to bound the Bessel functions in this way.)

Last edited by zetafunc (2016-10-09 04:08:16)

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#14 2016-10-09 04:10:25

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 108,752

Re: Sum of an integral involving Bessel functions

Can you write that up in Mathematica speak?


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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#15 2016-10-09 04:15:41

zetafunc
Member
Registered: 2014-05-21
Posts: 1,831
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Re: Sum of an integral involving Bessel functions

Here it is:

Sum[(Integrate[((x^2 + y^2)^(-3/4))*(((b - x)^2 + (c - y)^2)^(-3/
         4)), {x, 0, 2*Pi}, {y, 0, 2*Pi}]^2), {b, 1, Infinity}, {c, 1,
   Infinity}]

If it is possible to show this converges then we are done.

Last edited by zetafunc (2016-10-09 04:23:06)

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#16 2016-10-09 04:21:01

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 108,752

Re: Sum of an integral involving Bessel functions

Bracket missing.


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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#17 2016-10-09 04:23:15

zetafunc
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Registered: 2014-05-21
Posts: 1,831
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Re: Sum of an integral involving Bessel functions

Sorry, fixed it again.

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#18 2016-10-09 04:25:15

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 108,752

Re: Sum of an integral involving Bessel functions

Let me see what can be done with that now. Please hold on.


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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#19 2016-10-09 04:28:08

zetafunc
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Registered: 2014-05-21
Posts: 1,831
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Re: Sum of an integral involving Bessel functions

OK, thanks. I am currently waiting to see what Mathematica does with the original sum if b and c vary from 1 to 100.

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#20 2016-10-09 04:29:25

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 108,752

Re: Sum of an integral involving Bessel functions

Are there any singularities in that integral?


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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#21 2016-10-09 04:31:40

zetafunc
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Registered: 2014-05-21
Posts: 1,831
Website

Re: Sum of an integral involving Bessel functions

You mean the one in post #13? There is definitely a singularity at (x,y) = (0,0). Others may occur too if at any point (b,c) = (x,y). For the integral by itself though, there should only be a singularity at (0,0).

Last edited by zetafunc (2016-10-09 04:32:01)

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#22 2016-10-09 04:37:57

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 108,752

Re: Sum of an integral involving Bessel functions

A singularity at (0,0) is one of the endpoints of the integral can be a big problem. Is it a removable singularity?

There will of course be chances for more singularities at (b,c) = (x,y) as you point out.


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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#23 2016-10-09 04:39:58

zetafunc
Member
Registered: 2014-05-21
Posts: 1,831
Website

Re: Sum of an integral involving Bessel functions

Hmm, I don't think so. The limit as (x,y) tends to (0,0) blows up to infinity. That is a big problem. I may need to talk to my supervisor about this.

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#24 2016-10-09 04:42:09

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 108,752

Re: Sum of an integral involving Bessel functions

It is not impossible but usually singularities unless they are removable  cause integrals to equal infinity, in other words they do not converge and therefore do not exist.


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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#25 2016-10-09 04:44:22

zetafunc
Member
Registered: 2014-05-21
Posts: 1,831
Website

Re: Sum of an integral involving Bessel functions

The only way out of this that I can see would be to try to use the asymptotic expansions of the Bessel functions so that we end up with something with positive powers rather than negative ones.

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