Actually, Daniel, I highly appreciate your answer but... I found another formula on how to get that B. I don't know how much you know about physics but I remembered the Biot-Savart law. And I think that professor wants us to use it.

So, I hope you didn't waste to much time writing your last post, because I still need your help. :D

I have this vector:

In[13]:= tok = Simplify[sigma*Elpolje]

Out[13]= {(
 4 a A sigma (a^2 - x^2 + y^2))/((a^2 - 2 a x + x^2 + y^2) (a^2 + 
    2 a x + x^2 + y^2)), -((
  8 a A sigma x y)/((a^2 - 2 a x + x^2 + y^2) (a^2 + 2 a x + x^2 + 
     y^2)))}

And now I want to do a horrible integral. (If the integral simply can't be done, than I will of course do it with your method). The integral should be something like:

Integrate[(1/(4*
      Pi))*((tok[[2]]*z0)/((x0 - x)^2 + (y0 - y)^2 + z0^2)^(3/
       2)), {x, -Infinity, Infinity}, {y, -Infinity, Infinity}]

But.... mathematica can't do it. And now, since my knowledge of mathematica is really poor, I am asking you if I am doing something wrogn or is this really a bit too complicated for mathematica?

The problem is similar to finding the vector potential A corresponding to a magnetic field B, where B = curl A. The result is usually not unique, but may be narrowed down with the appropriate boundary conditions. I don't think Mathematica can directly solve vector differential equations like yours. You might try reformulating the problem as a set of simultaneous partial differential equations.