Author 
Comment/Response 
Bill Simpson

10/08/12 6:22pm
First problem I see, you are using {} where it should be () here r[x, y]^{λ[[n]]} and here r[x, y]^{λ[[n]]}. Actually you don't need {} or () for either of those, but () are acceptable.
Next, a generally useful thing. If you are creating a Table and it seems to be taking forever then try something like this
For[n = 1, n ≤ Nmax, n++,
Print["Starting ", n];
<insert your (1/(r[x,y]... expression>
]
and run that to see if it is making progress. You can also stick a Print[n]; inside your Table[] and have that give you some indication of progress, but you need to know where to put that and how to use the semicolon so that it does not interfere with your Table construction.
Sometimes a calculation will work for n==1,2,3 and then hang up taking forever when n==6 or n==10. But it appears when I do this it appears that after removing those pairs of {} that it is making some progress.
But, at least on my system the kernel crashed after a minute or so when I reduced Nmax to 5 and just tried to create your B table.
When I reduce Nmax to 2 I get this for B
In[9]:= B
Out[9]= {
Integrate[ArcTan[x, y]* Sin[ArcTan[x, y]], {x, 0, 1}, {y, 0, 1}]/(Pi*Sqrt[x^2 + y^2]),
Integrate[ArcTan[x, y]* Sin[3*ArcTan[x, y]], {x, 0, 1}, {y, 0, 1}]/(Pi*(x^2 + y^2)^(3/2))
}
So it does not appear that, at least my version, Mathematica can find a nice simple closed form solution for any of your integrals.
That will probably mean that ContourPlot may need to evaluate each of these integrals dozens or hundreds of times and that may be part of the reason it is taking a very long time on your system.
Mathematica is FANATIC about correct use of [] versus () versus {} and correct capitalization and correct use of = versus := versus == and there is even an === and those are all completely different. Make any tiny mistake with any one of those and you will get incorrect results or error messages you may not understand or even get nothing at all. I recommend using magnification on your notebook so that any tiny mistakes are easier to spot before you try running the notebook.
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