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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 ?**LearnMathsFree: Videos on various topics.New: Integration Problem | Adding FractionsPopular: Continued Fractions | Metric Spaces | Duality**

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

Hi;

Can you please show me your M code?

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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}]
```

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

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

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

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

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
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That is a very tough problem and may not have a closed form. What kind of answer are you looking for?

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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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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
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Perhaps I can get something out of this. First, I would like to test empirically your assertion that it converges.

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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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**bobbym****bumpkin**- From: Bumpkinland
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One problem is that I do not know what k is. Can you say something about k?

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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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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
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So, you want k to be another free variable or I am hoping we can at least bound it...

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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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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
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Can you write that up in Mathematica speak?

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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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**bobbym****bumpkin**- From: Bumpkinland
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Bracket missing.

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Sorry, fixed it again.

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**bobbym****bumpkin**- From: Bumpkinland
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Let me see what can be done with that now. Please hold on.

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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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**bobbym****bumpkin**- From: Bumpkinland
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Are there any singularities in that integral?

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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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**bobbym****bumpkin**- From: Bumpkinland
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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.

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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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**bobbym****bumpkin**- From: Bumpkinland
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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.

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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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