You appear to have left out the code that initializes Z_1 and Z_2. Depending how you initialize them, Z_1 and Z_2 could have more elements than X and Y.
To ensure Z_1 and Z_2 are the size you expect, try the following.
Z_1 = [];
Z_2 = [];
for i=1:1:100
for j=1:1:100
Z_sigma_1 = ((1 + X(i)) * sqrt(1 + X(i)^2)) / (2 * sqrt(2) * X(i) * Y(j));
Z_sigma_2 = ((X(i) - 1) * sqrt(1 + X(i)^2)) / (2 * sqrt(2) * X(i) * Y(j));
Z_1 = [Z_1 Z_sigma_1];
Z_2 = [Z_2 Z_sigma_2];
end
end
