Return analytic expression

Question

I would like to evaluate this summation:

\begin{align} \sum_{j=1}^{NM}\left[(u_{j_x} + u_{j_y})^2 - 2(1-\rho) u_{j_x} u_{j_y}\right], \end{align}

where $NM$ is some number greater than or equal to $N$, $u_{j_x} = \cos[\theta]\mu_{j_x} - \sin[\theta]\mu_{j_y}$ and $u_{j_y} = \sin[\theta]\mu_{j_x} + \cos[\theta]\mu_{j_y}$ and with

\begin{split} \mu_{j_x} &= \left[(j - 1)\%N - \left(\frac{N-1}{2}\right)\right] d_x, \\ \mu_{j_y} &= \left[\left\lceil{\frac{j}{N}}\right\rceil+ \left(\frac{N+1}{2}\right)\right] d_y. \end{split}

I have plot the result in Mathematica and have seen that the first term in the summation is a constant term and the second term oscillates. I would like to get some analytic expressions for the sum rather than simulating it numerically. The script I refer to is the following:

rotation = {};
rotation = Append[rotation, {"Theta", "sumterm"}];
μx[i_, n_, dx_] := (Mod[i - 1, n] + 1 - (n + 1)/2) dx; 
μy[i_, n_, dy_] := (Ceiling[i/n] - (n + 1)/2) dy;
μxn[i_, n_, dx_, dy_, θ_] := Cos[θ] μx[i, n, dx] - Sin[θ] μy[i, n, dy];
μyn[i_, n_, dx_, dy_, θ_] := Sin[θ] μx[i, n, dx] + Cos[θ] μy[i, n, dy];
n = 10; 
m = 10; 
s = m n;  
dx = 1;
dy = 1;
ρ = 0.0; 
Do[
  sumterm = 
    Sum[
      (μxn[i, n, dx, dy, θ] + μyn[i, n, dx, dy, θ])^2 - 
      2 (1- ρ) μxn[i, n, dx, dy, θ] μyn[i, n, dx, dy, θ], 
      {i, 1, s, 1}];
  rotation = Append[rotation, {θ // N, sumterm}], 
  {θ, 0 Degree , 360 Degree, 2 Degree}];
ListLinePlot[rotation, PlotRange -> All]

Answer 1

μx[i_, n_, dx_] := (Mod[i - 1, n] + 1 - (n + 1)/2) dx;
μy[i_, n_, dy_] := (Ceiling[i/n] - (n + 1)/2) dy;
μxn[i_, n_, dx_, dy_, θ_] := 
  Cos[θ] μx[i, n, dx] - Sin[θ] μy[i, n, dy];
μyn[i_, n_, dx_, dy_, θ_] := 
  Sin[θ] μx[i, n, dx] + Cos[θ] μy[i, n, dy];
n = 10;
m = 10;
s = m n;
dx = 1;
dy = 1;
ρ = 0;

With your code, your sumterm is a constant, i.e., independent of θ

sumterm = Sum[(μxn[i, n, dx, dy, θ] + μyn[i, n, dx, dy, θ])^2 - 
    2 (1 - ρ) μxn[i, n, dx, dy, θ] μyn[i, n, dx, dy, θ], 
      {i, 1, s, 1}] // Simplify

(* 1650 *)

It is also independent of ρ

Clear[ρ]

sumterm = Sum[(μxn[i, n, dx, dy, θ] + μyn[i, n, dx, dy, θ])^2 - 
    2 (1 - ρ) μxn[i, n, dx, dy, θ] μyn[i, n, dx, dy, θ], 
      {i, 1, s, 1}] // Simplify

(* 1650 *)