Is it possible to evaluate this double summation which includes an integral with Mathematica for some given $r$?
$$\sum_{m=-\infty, m\neq 0}^\infty\sum_{n=-\infty, n\neq 0}^\infty m \int_0^{2 \pi} \log [2r - r\cos \theta + m(\cos \theta-1) + n \sin\theta + m^2 + n^2] d\theta$$
Your sum doen't converge at all. Look how your summands behave if you go to infinity with series. Here the m-sum, postivie and negative added:
serm1 =
Series[m Log[
2 r - r Cos[θ] + m (Cos[θ] - 1) +
n Sin[θ] + m^2 + n^2] -
m Log[2 r - r Cos[θ] - m (Cos[θ] - 1) +
n Sin[θ] + m^2 + n^2], {m, Infinity, 1}] // Normal
(* -2 + 2 Cos[\[Theta]] *)
Every step you add a constant value, therefore your sum must go to infinity. It's even worse with the n-Sum:
sern1 =
Series[m Log[
2 r - r Cos[θ] + m (Cos[θ] - 1) +
n Sin[θ] + m^2 + n^2] +
m Log[2 r - r Cos[θ] + m (Cos[θ] - 1) -
n Sin[θ] + m^2 + n^2], {n, Infinity, 1}] // Normal
(* -4 m Log[1/n] *)
Limit[sern1, n -> Infinity]
(* m \[Infinity] *)
You can calculate a partial sum numericaly (analytical integration didn't work) to have an impression it goes to infinity:
nint[r_, m_, n_] :=
NIntegrate[
Log[2 r - r Cos[θ] + m (Cos[θ] - 1) +
n Sin[θ] + m^2 + n^2], {θ, 0, 2 Pi}]
nsum[r_, m0_, n0_] :=
NSum[m nint[r, m, n] + m nint[r, m, -n], {n, 1, n0}, {m, -m0, m0}]
Table[nsum[1, k, k], {k, 1, 10}]
)* {-6.10232, -34.2553, -88.7863, -169.822, -277.058, -410.236, \
-569.171, -753.729, -963.815, -1199.36} *)
