```
NIntegrate[(BesselJ[1, x]*BesselJ[1, b - x])/(Abs[x]*
Abs[b - x]), {x, -Infinity, Infinity}]
```

Some results:

```
b = 1: -0.307306
b = 2: 0.151302
b = 4: 0.29021
b = 8: -0.0672058
b = 16: 0.0372027
b = 32: -0.0154359
b = 64: -0.00547838
b = 128: -0.000120145
b = 256: 0.00056718
b = 512: 0.000169772
b = 1024: -0.0000548278
b = 2048: -0.0000149542
```

and plug in values of b to investigate how the value of the integral changes. However, I'm not sure if this simplification tells us something useful about the original problem -- though this is very easy to integrate with M and I have some results for how the integral changes as b varies between powers of 10.

]]>Alternatively, we could instead get rid of the outer integral and instead figure out what the dependence on b,c is.

How are you going to do that?

]]>I am getting an error almost as big as the answer with other messages about slow convergence. There is no reliability in that answer at all.

We have talked quite a bit about lots of stuff. I do not care about your physical appearance, your gender, your religion, your social status or how much money you have. I only know you from your writings, so in the truest sense I consider you a friend. So as your friend I would like to offer some advice:

Here is a quote from someone over at Mathematica.SE for an integral that is not as complex as yours.

Felix wrote:

You have lots of stuff in the integral that does not depend on x. You should first manually reduce the complexity of the integrand and the number of variables to the minimum possible. And then you need to use assumptions for all remaining parameters since the integral most likely does not converge for arbitrary complex values of the parameters. Even then it might not work, but in the way you write it even much simpler integrals would not give a result.

It is the job of the human to prepare the problem for M to solve. That is our job now. Mathematica can not do the impossible. I have watched you suffering over these type of multiple integrals for months with little progress being made. I know you can make progress when the problem allows it. This type of problem is not amenable to progress either by you or M or anybody on any forum we have tried. It does look like this is a dead end.

Now I ask you, can you shorten that to no more than a double integral with very few unrelated parameters? These are the types M can do, these are the only types anyone can do. Unless your supervisor is willing to give out a lot more information about this problem then maybe it is time to abandon this and look for something else. What have you got to lose? Please think about this.

]]>```
NIntegrate[(BesselJ[1, Sqrt[x^2 + y^2]]*
BesselJ[1, Sqrt[z^2 + w^2]] BesselJ[1,
Sqrt[(b - x)^2 + (c - y)^2]]*
BesselJ[1, Sqrt[(b + z)^2 + (c + w)^2]])/(Sqrt[x^2 + y^2]*
Sqrt[z^2 + w^2]*Sqrt[(b - x)^2 + (c - y)^2]*
Sqrt[(b + z)^2 + (c + w)^2]), {x, 0, 1000000}, {y, 0,
1000000}, {b, 0, 1000000}, {c, 0, 1000000}, {z, 0,
1000000}, {w, 0, 1000000}]
```

I've managed to do the computation up to 10^21. The trouble is, the error is huge (about 20000). However, the error does get smaller if we integrate over a larger and larger region. For instance, integrating up to 10^30 gives -506 with an error of 5422. Up t0 10^100, we get 121.589 and 392588.3228022117 for the integral and error estimates.

]]>If the square causes a problem, there is a 6-integral representation also. I am currently trying this out with Mathematica. Rewriting it like this could help:

```
NIntegrate[(BesselJ[1, Sqrt[x^2 + y^2]]*
BesselJ[1, Sqrt[z^2 + w^2]] BesselJ[1,
Sqrt[(b - x)^2 + (c - y)^2]]*
BesselJ[1, Sqrt[(b + z)^2 + (c + w)^2]])/(Sqrt[x^2 + y^2]*
Sqrt[z^2 + w^2]*Sqrt[(b - x)^2 + (c - y)^2]*
Sqrt[(b + z)^2 + (c + w)^2]), {x, 0, 5}, {y, 0, 5}, {b, 0,
5}, {c, 0, 5}, {z, 0, 5}, {w, 0, 5}]
```

I am currently running that with those limits.

]]>