we consider

There will be singularities, but he says that they won't lead to divergence in this case. The integral

]]>I did a much easier one you can look at as an example over here:

]]>```
NSum[NIntegrate[(Cos[Sqrt[x^2 + y^2] - 3*\[Pi]/4]*
Cos[Sqrt[(b - x)^2 + (c - y)^2]])/((((x^2 + y^2)^(3/
4)))*((b - x)^2 + (c - y)^2)^(3/4)), {x, 1, 10}, {y, 1,
10}], {b, 1, 10}, {c, 1, 10}]
```

works a lot better and yielded 4.02468 in 16 seconds, surprisingly.

```
NSum[NIntegrate[(Cos[Sqrt[x^2 + y^2] - 3*\[Pi]/4]*
Cos[Sqrt[(b - x)^2 + (c - y)^2]])/((((x^2 + y^2)^(3/
4)))*((b - x)^2 + (c - y)^2)^(3/4)), {x, 1, 50}, {y, 1,
50}], {b, 1, 20}, {c, 1, 20}]
```

produced 3.72777 in 45.49 seconds.

```
NSum[NIntegrate[(Cos[Sqrt[x^2 + y^2] - 3*\[Pi]/4]*
Cos[Sqrt[(b - x)^2 + (c - y)^2]])/((((x^2 + y^2)^(3/
4)))*((b - x)^2 + (c - y)^2)^(3/4)), {x, 1, 500}, {y, 1,
500}], {b, 1, 20}, {c, 1, 20}]
```

produced 4.23585 in 198.92 seconds.

]]>My revised post only uses one NIntegrate and gets faster results.

]]>```
NIntegrate[
NIntegrate[(Cos[Sqrt[x^2 + y^2] - (3 \[Pi])/4] Cos[
Sqrt[(1 - x)^2 + (1 - y)^2]])/((x^2 + y^2)^(3/
4) ((1 - x)^2 + (1 - y)^2)^(3/4)), {x, 1, 1000}], {y, 1,
1000}]
```

produced 1.13796 in 89.72 seconds.

And here is what happens when trying a sum over one variable.

```
NSum[NIntegrate[
NIntegrate[(Cos[Sqrt[x^2 + y^2] - (3 \[Pi])/4] Cos[
Sqrt[(b - x)^2 + (10 - y)^2]])/((x^2 + y^2)^(3/
4) ((b - x)^2 + (10 - y)^2)^(3/4)), {x, 1, 1000}], {y, 1,
1000}], {b, 10, 12}]
```

produced 0.655201 in about 50 seconds. And that is with just 1 variable summed over 3 terms.

Currently running a double NSum with a double NIntegrate and that is producing lots and lots of errors, several of which include the words "catastrophic loss of precision".

EDIT:

```
NSum[NSum[
NIntegrate[
NIntegrate[(Cos[Sqrt[x^2 + y^2] - (3 \[Pi])/4] Cos[
Sqrt[(b - x)^2 + (c - y)^2]])/((x^2 + y^2)^(3/
4) ((b - x)^2 + (c - y)^2)^(3/4)), {x, 1, 10}], {y, 1,
10}], {b, 1, 10}], {c, 1, 10}]
```

produced 3.05475*10^16 in about 395 seconds.

]]>EDIT: Sorry, didn't notice your post just now. I managed to get your original code (with the nested NIntegrates) to work after closing Mathematica fully.

]]>```
NIntegrate[(Cos[Sqrt[x^2 + y^2] - 3*\[Pi]/4]*
Cos[Sqrt[(100 - x)^2 + (100 - y)^2]])/((((x^2 + y^2)^(3/
4)))*((100 - x)^2 + (100 - y)^2)^(3/4)), {x, 1, 5}, {y, 1,
5}]
```