Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫  π  -¹ ² ³ °

You are not logged in.

## #76 2016-11-02 10:08:16

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,249
Website

### Re: Sum of an integral involving Bessel functions

That is the sum of the modulus of the old integral squared, which unfortunately is now wrong, according to my supervisor. The new integral is in the newer thread (though the old thread was not made in vain as we still learned some things from it).

My supervisor also says he is sceptical that I can get the integral to converge for d > 2, because he says he remembers trying something similar himself and finding that the integral diverged.

Offline

## #77 2016-11-02 10:15:22

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

### Re: Sum of an integral involving Bessel functions

I would daresay that your supervisor and yourself are among the world's leading authorities on either double integral. I can not touch it with anything I know.

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.

Offline

## #78 2016-11-03 01:49:19

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,249
Website

### Re: Sum of an integral involving Bessel functions

I think I know the answer to my question on MO. I will post up the solution when I get home/can type on something other than my phone.

Offline

## #79 2016-11-03 04:19:03

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

### Re: Sum of an integral involving Bessel functions

Hi;

You can post it and accept 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.

Offline

## #80 2016-11-03 11:19:56

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,249
Website

### Re: Sum of an integral involving Bessel functions

I've just sent my solution to my supervisor -- about 3 pages long. I'll wait and see what he has to say about it.

The case for d = 2 certainly seems much more difficult. If I can't get an answer I might consider putting a bounty up on there, although it might hurt to lose a third of my reputation.

Offline

## #81 2016-11-03 13:04:21

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

### Re: Sum of an integral involving Bessel functions

How long will it take before you know?

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.

Offline

## #82 2016-11-03 18:49:18

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,249
Website

### Re: Sum of an integral involving Bessel functions

I'm not sure. He could respond over the weekend. It's a fairly simple proof, though -- I can post it in Members Only if you would like to have a look.

On the other hand I get the feeling the result might not be correct because of a more recent result.

Last edited by zetafunc (2016-11-03 21:45:21)

Offline

## #83 2016-11-03 23:15:36

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

### Re: Sum of an integral involving Bessel functions

On the other hand I get the feeling the result might not be correct because of a more recent result.

What is that result?

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.

Offline

## #84 2016-11-04 01:19:29

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,249
Website

### Re: Sum of an integral involving Bessel functions

There is a result which seems to suggest that the
-norm is bounded below by
. My results are saying they are bounded above by
which is obviously impossible. Then again, that more recent result would also seem to contradict some older results about the
and
-norms of the remainder, which suggests perhaps we are misinterpreting the results of that paper.

Offline

## #85 2016-11-04 05:31:12

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

### Re: Sum of an integral involving Bessel functions

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.

Offline

## #86 2016-11-04 05:32:35

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,249
Website

### Re: Sum of an integral involving Bessel functions

I want to ask him about it, but the next time I can see him will be next Friday. Although the author of that paper is at my uni also, so I suppose I could ask the author.

Offline

## #87 2016-11-04 05:37:56

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

### Re: Sum of an integral involving Bessel functions

That is a good idea. Start to set up a meeting to see him.

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.

Offline

## #88 2016-11-04 07:46:17

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,249
Website

### Re: Sum of an integral involving Bessel functions

Here is another post on MO for the case d = 2. The title is a bit simpler so hopefully it will attract some people, given that it is Friday.

Offline

## #89 2016-11-04 12:12:44

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

### Re: Sum of an integral involving Bessel functions

Does your integral look like Jacky thinks?

http://math.stackexchange.com/questions … 42#1993642

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.

Offline

## #90 2016-11-04 17:49:08

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,249
Website

### Re: Sum of an integral involving Bessel functions

For general d, yes.

Offline

## #91 2016-11-04 19:34:31

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

### Re: Sum of an integral involving Bessel functions

That ought to be doable with M, did you try it just to see?

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.

Offline

## #92 2016-11-04 21:25:16

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,249
Website

### Re: Sum of an integral involving Bessel functions

Yeah -- but I already posted this integral in the other thread. Couldn't get an answer with M, but it doesn't matter, because he proved it converges for d > 2. d = 2 is what I'm waiting on, though someone on MO is claiming it converges after bounding the Bessel functions. I don't believe their claim, however.

Some interesting observations: if b = c = 1, then the integral below grows very large. If b and c are large, then the integral grows more slowly and has a much smaller value when integrated over a large region (say, [-10000,10000]). This is similar to the observations my supervisor made on the annulus problem (that the lattice vectors had to be bounded away from zero).

Last edited by zetafunc (2016-11-05 03:48:14)

Offline

## #93 2016-11-05 05:02:46

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

### Re: Sum of an integral involving Bessel functions

though someone on MO is claiming it converges after bounding the Bessel functions.

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.

Offline

## #94 2016-11-05 05:03:24

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,249
Website

Offline

## #95 2016-11-05 05:18:51

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

### Re: Sum of an integral involving Bessel functions

I would not go with his answer just yet for the practical reason that he has not been able to deal with your comment yet. I would really like to see some other form for the integral than the one given in post 92.

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.

Offline

## #96 2016-11-05 05:29:57

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,249
Website

### Re: Sum of an integral involving Bessel functions

I agree. I'm mostly sceptical about this change to polar co-ordinates, because there is no way that integral he is talking about converges, unless I've misunderstood what he says.

He's just given an answer but I still don't understand it, I'll need to think about it for a while.

Offline

## #97 2016-11-05 05:56:16

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

### Re: Sum of an integral involving Bessel functions

Yea, I see it too. See you in a bit. Got a chore to do.

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.

Offline

## #98 2016-11-05 06:39:47

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,249
Website

### Re: Sum of an integral involving Bessel functions

OK, see you later.

M suggests that larger values of b tend to result in something very close to zero. Using M on the integral

we have:

``````NIntegrate[((x^2 + y^2)^(-1/2)) (((100 - x)^2 + (100 - y)^2)^(-1/2))*
BesselJ[1, (x^2 + y^2)^(1/2)]*
BesselJ[1, (((100 - x)^2 + (100 - y)^2)^(1/2))], {x, -10^5,
10^5}, {y, -10^5, 10^5}]``````

which produces -0.000411696 with an error of 0.002530016802142993. The output and error get much bigger though for a range higher than [-10^6, 10^6]. I've tried plotting this to see how NIntegrate behaves for varying b, or for a varying range, but I can't get either to work.

Offline

## #99 2016-11-05 08:04:42

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

### Re: Sum of an integral involving Bessel functions

Hi;

I am getting a lot of error messages with that so I would not be too confident in the answer or that estimate. Do you use v11?

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.

Offline

## #100 2016-11-05 08:06:42

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,249
Website

### Re: Sum of an integral involving Bessel functions

I'm using 10.3 at the moment, and my licence expires in about 10 days.

Offline